STUDY OF THE EFFECT OF SHAPE ON THE AERODYNAMIC PERFORMANCE OF HYPERLOOP POD
By ANUP JAIN
A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2019 © 2019 Anup Jain ACKNOWLEDGMENTS I would like to express my gratitude towards my supervisor, Dr. Bhavani Sankar, for his valuable guidance, help, and support. Without him, this project would not have been possible to be realized. It was great to have someone like him who was always available whenever I had any questions regarding the thesis. I want to thank my Co-chair, Dr. Ashok Kumar, for his valuable suggestions at every stage of the project and for all the discussions we had that helped me tremendously. I want to thank Dr. Siddharth Thakur for his guidance and consultation with the entire CFD process during the thesis. My special thanks, to the Support Engineers at Friendship Systems, for providing me free access to CAESES software and answering all my questions regarding it. Lastly, I want to thank all the people who directly or indirectly contributed to the successful completion of the Thesis
3 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...... 3 LIST OF TABLES ...... 6 LIST OF FIGURES ...... 7 LIST OF DEFINITIONS ...... 8 ABSTRACT ...... 10
CHAPTER 1 INTRODUCTION ...... 12 1.1 Objective ...... 12 1.2 Limitations ...... 13 1.3 Thesis Outline ...... 13 2 LITERATURE REVIEW ...... 14 2.1 Background ...... 14 2.2 Flow Field ...... 15 2.3 Design Parameters and its effect on Drag ...... 17 2.4 Preliminary Shape Analysis ...... 18 2.5 CFD Introduction ...... 19 2.6 Turbulence Modeling ...... 21 3 METHOD ...... 24 3.1 Geometry ...... 24 3.2 Mesh ...... 26 3.3 Numerical Solver ...... 28 3.3.1 Finite Volume Method ...... 28 3.3.2 Pressure Based Solver and Density Based Solver ...... 29 3.4 Numerical Model ...... 30 3.5 Boundary Conditions ...... 31 3.6 K-omega SST Turbulence Model ...... 32 3.7 Convergence and Verification ...... 34 4 SURROGATE MODELING ...... 39 4.1 Polynomial Regression Model ...... 40 4.2 Kriging Regression Model ...... 41 4.3 Surrogate Results ...... 42
4 5 RESULTS ...... 45 5.1 Effect of Shape of Head ...... 45 5.2 Effect of Shape of Tail ...... 50 5.3 Sensitivity Analysis ...... 55 6 CONCLUSION ...... 58 APPENDIX:ICEM CFD SCRIPT ...... 59 REFERENCES ...... 69 BIOGRAPHICAL SKETCH ...... 72
5 LIST OF TABLES Table page 3-1 Coordinates of the Control Points of the B-Spline ...... 26 3-2 Constraint of the variables of the control points ...... 27 4-1 Goodness of Fit for Polynomial and Kriging Regression Model ...... 43 5-1 Maximum Pressure Generated on Head for different Tail shapes ...... 52
6 LIST OF FIGURES Figure page 3-1 Parametric Pod Geometry ...... 26 3-2 The Structured Grid around Pod with Semicircle head and Blunt tail ...... 28 3-3 Effect of Head shape on the drag for different operating pressures and velocity ... 34 3-4 Residuals ...... 36 3-5 Net Mass Flow Rate (across Inlet and Outlet) Vs Iterations ...... 36 3-6 Plot of Drag Coefficient Vs Iterations ...... 37 3-7 Plot of Drag Vs Iterations ...... 37 4-1 Plot of Drag obtained from the Response Surface vs Actual Observed Values. .... 43 5-1 Effect of Head shapes on drag for different Blockage ratio ...... 45 5-2 Effect of Head shapes on drag for different operating Pressures ...... 46 5-3 Distribution of Pressure Coefficient around the nose for different Head shapes ... 47 5-4 Pressure Distribution across the Semicircle shaped head ...... 48 5-5 Pressure Distribution across the Arched shaped head ...... 49 5-6 Pressure Distribution across the Splined Elliptic shaped head ...... 50 5-7 Effect of Head shape on the drag for different operating pressures and velocity ... 51 5-8 Effect of Tail shapes on drag for different Blockage ratio ...... 51 5-9 Maximum Pressure generated on the head due to different shapes of Tail ...... 52 5-10 Velocity Distribution across the pod with Blunt Tail ...... 53 5-11 Velocity Distribution across the pod with Elliptic Tail ...... 53 5-12 Velocity Distribution across the pod with Splined shaped Tail ...... 54 5-13 Effect of Head shape on the drag for different operating pressures and velocity ... 54 5-14 Effect of Tail shape on Drag for different operating Pressures ...... 55 5-15 Effect of Tail shape on Drag for different Velocity ...... 56 5-16 Sensitivity based on Polynomial Regression Model ...... 56 5-17 Sensitivity based on Kriging Regression Model ...... 57
7 LIST OF DEFINITIONS
2D Two-Dimensional 3D Three-Dimensional CFD Computational Fluid Dynamics COD Coefficient of Determination DBS Density Based Solver DNS Direct Numerical Simulation DOE Design of Experiment FVM Finite Volume Method LHS Latin Hypercube Sampling Maglev Magnetic levitation NURBS Non-Uniform Rational Basis Spline OSF Optimal Space Filling PBS Pressure Based Solver PRG Polynomial Regression Model RANS Reynolds-average Navier-Stokes equations SST Shear Stress Transport i ith vector element j jth vector element a Acceleration β Blockage ratio
Kb Boltzmann constant ρ Density µ Dynamic viscosity
νT Eddy viscosity R Gas constant g Gravitational constant
8 q Heat flux vector δ Kronecker delta < Less than M Mach number λ Molecular mean free path << Much smaller than y+ Non-dimensional wall distance ∼ Of the order of ∂ Partial derivative p Pressure
Cp Pressure coefficient
Re Reynolds number ω Specific turbulence dissipation rate T Temperature k Turbulence kinetic energy ▽ Vector differential operator − Vector notation
9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science STUDY OF THE EFFECT OF SHAPE ON THE AERODYNAMIC PERFORMANCE OF HYPERLOOP POD By Anup Jain December 2019 Chair: Bhavani V. Sankar Major: Mechanical Engineering The Hyperloop is a new form of a ground transportation system that consists of pods carrying passengers or cargo traveling through a partially evacuated tube. Hyperloop, in a nutshell, is about eliminating two factors that slow down traditional vehicles, friction, and air resistance. To tackle the former, we levitate the pod using electromagnets and propel it using linear induction motor, whereas to address the later, we remove the air inside the tube such that we have a low-pressure environment. However, this motion of the hyperloop pod through a confined space, at transonic speeds imposes unique fluid dynamic challenges such aschoked flow, Kantrowitz Limit. These factors have a substantial impact on the performance ofthe vehicle and hence investigating the aerodynamics becomes of paramount importance while analyzing such systems. This master’s thesis aims to investigate the effect of the different head and tail shapeson the aerodynamic drag under varying blockage ratios, pressure, and velocity. We perform 2D steady-state simulations on three distinct heads and tail shapes, respectively. The blockage ratio differs from 0.25 - 0.6, pressure ranges from 10 Pa to 100 KPa, and velocity changes from 50 m/s to 300 m/s. The shape of the head is found to have no pronounced effect on the drag for any value of the blockage ratio. On the contrary, it is the shape of the tail that has a significant impact on the aerodynamic drag regardless of the pressure, velocity, and blockage ratio. For a given value of pressure in the tube, there is a limiting velocity above which the
10 shape of the head has a substantial effect on the aerodynamic drag. The lower the pressure in the tube, the higher the limiting velocity and viceversa. Sensitivity analysis is performed to verify the accuracy of the results,on 2D parametric geometry constructed using B-splines. Two surrogate models, the second-order polynomial and Kriging are used to compute the sensitivities. The results obtained are in line with our previous findings and confirm that the aerodynamic drag is more sensitive to the shape of thetailthan to the shape of the head.
11 CHAPTER 1 INTRODUCTION Hyperloop is a new form of transportation method that consists of passenger pods traveling through a partially evacuated tube. The pressure inside the tube is about 0.1%- 1% SpaceX (2013) of the atmospheric pressure obtained by removing the air inside the tube. The pods are levitated using electromagnets, and the propulsion is handled using linear induction motors. Elon Musk and the team of engineers from Tesla and Space Exploration Technologies initially proposed this idea in an alpha white paper SpaceX (2013) in August 2013 . The Hyperloop system merges a combination of ground transportation and aerospace technology to achieve new capability. Because of the partial vacuum within the tube and no contact with the ground. The hyperloop pod faces a little to no resistance as it moves. Such an environment enables pods to move at airliner speed on the ground using very little energy. Hence, the pod can spend more time traveling at cruise conditions in contrast to an airliner which has inevitable inefficiencies accompanied with it (taxi time, climbing, descent, holding time, etc.). This gives the hyperloop system a unique advantage over air travel for a short-haul journey. Despite the low pressure in the tube, and hence the low density of air, the hyperloop pod still experiences an aerodynamic drag. Improving the performance characteristics of the hyperloop pod, such as traveling at extraordinary speeds with a minimum amount of energy consumption, entails the need for careful aerodynamic design. The study of Hyperloop brings new sets of fluid dynamic challenges because of its motion through a confined space which makes it an internal aerodynamic problem as opposed to classical external aerodynamics problem found in automobiles, trains, and airplanes. 1.1 Objective
The primary purpose of this master’s thesis is to study the effect of different shapes on head and tail on the aerodynamic drag under varying blockage ratios, pressure, and velocity. The aim is also to obtain a better knowledge of the physical phenomena and the key
12 parameters and shapes responsible for the drag contribution on the surface of the pod and to get a better picture of how changes in those shapes will help to gain the most drag reduction. 1.2 Limitations
Studying the effect of different shapes on drag under varying conditions will require many CFD simulations. Moreover, the generation of the response surface to calculate the sensitivity also demands much computational effort. Considering the availability of time, computational power, and memory at hand, we performed the CFD analysis on 2D geometry, assuming steady-state conditions. Since the amount of CPU time and memory needed is directly proportional to the grid points present in the mesh, we will restrict the mesh in this case to approximate 0.6 million cells (upon grid convergence study). 1.3 Thesis Outline
The thesis report is organized into eight chapters. Chapter 1 provides general information about the Hyperloop system. It also introduces the aim of the study, along with the justification of the scope and the approach used. Chapter 2 provides a literature review and summarizes the current state of the knowledge and research related to the hyperloop system. It also reviews the fluid flow regime, preliminary design parameters, primary shape analysis, and theoretical concepts in CFD and turbulence modeling. In Chapter 3, the CFD methodology is described. The steps involved in the analysis like geometry, mesh, solver settings, boundary conditions, convergence criteria, and the grid convergence study is presented. Chapter 4 provides a short review of the surrogate models and the design of experiments used in this work. Results obtained from the study, are presented discussed in Chapter 5 and conclusions drawn from them are discussed in Chapter 6. Finally, Chapter 7 discusses the recommendation for future work.
13 CHAPTER 2 LITERATURE REVIEW 2.1 Background
The idea of moving people and goods by pneumatic tubes has been around for quite some time. It was first proposed way back in 1799 when George Medhurst took out a patentto transport goods by an airtight metal tube. Through the 1800s, there were several attempts at implementing what commonly became known as atmospheric railways, which included anything that used differential air pressure to provide power for propulsion. The Isambard Kingdom Brunel’s atmospheric railway was the first of its kind to run in1847. Throughout mid-1850s several atmospheric railways were built in Dublin, London, and Paris. Fast forward to 1910, an American rocket pioneer Robert Goddard the first proposed the concept of vacuum trains. In 1972 Rand Corporation conceived the idea of Very High Speed Transit System (VHST) Salter (1972) for transportation that involved electromagnetically levitated and propelled cars traveling through evacuated tunnels. It is this reduced pressure or vacuum propelled transportation apparatus that seems to have the most in common with the modern hyperloop. Back in 2013, Tesla and SpaceX founder Elon Musk first published about the modern-day hyperloop in an alpha white paper SpaceX (2013). According to the white paper, the hyperloop capsule travels at a speed of 700 miles per hour within an evacuated tube maintained at a pressure of 100 Pa SpaceX (2013). They levitate the pod using air bearings. They have avoided the use of a conventional wheel and axle system because of friction losses and dynamic stability at transonic speeds. The pod is propelled using a linear electric motor instead of a magnetic levitation because of a high construction and operation cost. The white paper also introduces a compressor on the front of the pod. According to the paper, it serves two important purposes. First, it increases the aerodynamic efficiency of the pod. When the pod is traveling at transonic speeds, the flow around the pod gets choked due to sonic conditions known as Kantrowitz limit. The air gets collected in front of the pod, exerting pressure on it. The on board compressor helps to bypass this incoming flow of air, thereby
14 reducing the pressure and eventually the pressure drag. Secondly, it acts as a supplier of compressed air to the air bearings required for levitation. This impressive idea behind the hyperloop and the potential to link the cities directly sparked intense interest among the people all around the world. This led to the formation of several start ups and student teams developing various aspects of the hyperloop technology with varying degree of success. Now several fully-fledged companies are making significant efforts to bring the hyperloop system into reality. 2.2 Flow Field
The hyperloop pod travels in an unusual flow regime, and hence the prior knowledge of the flow field model is necessary. Since the operating pressure inside the tube is low,oneneeds to know whether the continuum approach is suitable to model the flow field. The Knudsen number Knudsen and Partington (1934) is one such dimensionless parameter used to describe the boundary of continuum flow. It is the ratio of the mean free path λ to a characteristic length scale L of the body of influence. Mean free path is the average distance traveled bythe fluid molecule before encountering collision event with the second molecule. It is givenas,
λ K = n L
For Boltzmann gas it can be determined as,
KbT Kn = √ 2πd2pL
If the characteristic length of the body of influence is higher than λ, i.e. Kn ≪ 1 there will be more molecule to molecule collisions and flow will appear to this body as a continuous substance. On the other hand, if the λ is comparable to L, or greater than L, then each collision will be felt by the body. In such a case, the fluid cannot be considered as a continuum and needs to has to be dealt with the statistical approach.In our case, the Knudsen number is of the order of 10−5, which is far below the continuum limit, implying that we can model the flow as the continuum.
15 Another challenge that stems from the lower internal tube pressure is the relatively low Reynolds number. The Reynolds number depends on the density of the fluid which directly depends on the pressure. Since the pressure inside the tube is low, the density is also low and hence the Reynolds number. The Reynolds number for the flow is around 105, which implies that the flow will transition from laminar to turbulent on the surface of the pod.Since both the laminar and turbulent flow yield different performance characteristics, capturing the transition is essential. Furthermore, the low Reynolds number enhances the risk of early separation of the boundary layer, which results in an increase in pressure drag on the pod. Finally, the motion of the hyperloop pod at transonic speeds through a confined space in the tube imposes other sets of challenges known as Kantrowitz limit as discussed below. Kantrowitz limit: Kantrowitz limit refers to the physical phenomenon of choked flow when the local flow reaches sonic conditions. If the pod is traveling at high subsonic speeds, the air in the tube needs to accelerate when it passes through the small cross-sectional area between the pod and the tube to satisfy the continuity principle. As a result, sonic conditions are easily reached in the bypass region.If the pod speeds up further, the column of the air in front of the pod starts building up pressure, since the mass flow rate that can pass around the pod is limited due to choking thereby regarding its motion. Choked flow is a limiting condition where the mass flow cannot increase even if the downstream pressure is decreased. Ithappens when the fluid is too dense or travels too fast through a minimal area. So if the hyperlooppod goes too fast, the air cannot move around it fast enough, and it will start pushing a column of air in front of it, which will slow it down. One solution is to put a compressor and move that mass of air to the rear of the pod so that air pressure will not slow the pod. The hyperloop white alpha paper uses this approach. Another solution is to increase the throat area, i.e., minimizing the blockage ratio. However, neither of the solutions is perfect. First, minimizing the blockage ratio implies either increasing the diameter of the tube, which increases the construction cost, or reducing the pod size, which reduces the payload carrying capacity. Second, the use of the transonic compressors at such a low Reynolds number will require a lot
16 of research and development. Since they are not commonly used in any aerospace application today Opgenoord and Caplan (2018). 2.3 Design Parameters and its effect on Drag
To improve the aerodynamic performance and the efficiency of the hyperloop pod, itis necessary to identify and understand the key parameters and the phenomenon responsible for the generation of aerodynamic drag. Generally, there are two types of drag associated with the vehicle in conventional ground transportation, pressure drag, and friction drag. The pressure drag is caused by the separation of flow due to an adverse pressure gradient ,and the formation of the wake while the viscous effect of the fluid causes the friction drag. However, for the hyperloop system, an additional form of drag exists, called wave drag, due to the formation of shock waves. The shock waves are formed when the local flow velocity is higher than the speed of the sound. These shock waves act as a discontinuity in the flow and create an abrupt variation in density, pressure, and temperature. Since the hyperloop pod is intended to travel at transonic to supersonic speeds, the formation of the shock waves is prevalent. Because of the shock waves, the boundary layer separates at the foot of the shock, leading to the formation of the wake region. Consequently, the momentum deficit in the wake results in a significant amount of pressure drag. Hence, the presence of the shock wavesis undesired and should be avoided. The pressure drag and the friction drag are directly proportional to the square of the operating velocity of the vehicle. Hence, with the increase in velocity, the pressure drag and the friction drag increase significantly. Besides the operating velocity of the vehicle, the pressure drug is linearly proportional to the density of the fluid. Since we deal with an ideal gas, the density of the fluid is governed by pressure. Hence, the pressure drag increases linearly with the increase in pressure. Similarly, the friction drag is linearly proportional to the area of the surface in contact with the fluid. Increasing the length of the pod will also increase the friction drag. Moreover, both the drags depend on the shape and texture of the vehicle.
17 It is found that for a relatively smaller vehicle, the total drag mainly comprises of pressure drag while for longer vehicles, the total drag is dominated by friction drag. However, this is exactly opposite in case of tube train (which does resemble a hyperloop system). Despite the low pressure in the tube, the pressure drag constitutes the higher percentage of the total aerodynamic drag. drag. This can be attributed to the pressure exerted by the compressed air in front of the train when the train is moving through the tube. Kim et al. (2011) found that for a tube train system the ratio of pressure drag to viscous drag is approximately 10:1 for a blockage ratio of 0.25, which increases to 20:1 when the blockage ratio becomes 0.75. Thus not only shows how dominating the pressure drag is as compared to the viscous drag but also shows the high dependence of the drag on the blockage ratio. E.g., at a blockage ratio of 0.5 and an operating speed of M= 0.57, shock waves were observed at the rear of the vacuum train Kim et al. (2011). However, when the blockage ratio was reduced to 0.25, for the same speed, no shock waves were observed. The blockage ratio does have a significant indirect impact on the pressure drag and the wave drag. Furthermore, the operating velocity is inversely proportional to the blockage ratio and internal pressure in the tube. Hence, to improve aerodynamic efficiency, the blockage ratio and the internal tube pressure should be kept to minimum. However, minimizing the blockage ratio negatively impacts the operating cost since it would require tubes of larger diameter. The blockage ratio is limited by the maximum size of the tube and the minimum size of the pod. It was found that the blockage ratio should be between 0.25 to 0.7, for consistent aerodynamic performance Zhang (2012). In a nutshell, the internal pressure in the tube, the blockage ratio, the operating velocity of the vehicle, and the length of the vehicle affect the aerodynamic drag. 2.4 Preliminary Shape Analysis
Out of the total aerodynamic drag experienced by the vehicle, pressure drug accounts for the most. The shape of the vehicle, most importantly, the head and the tail, drives the pressure drag. The research about the aerodynamic analysis and the performance of the tube train system dates back to early 1970. In a study conducted by the office of high-speed
18 ground transportation, United States (*reference TU delft 18) it was found that the train with semicircular shaped head and tail had a drag coefficient ranging between 0.015 to 0.11, for Reynolds number of the order of 105 and the blockage ratio ranging between 0.125 to 0.5, that is comparable to the modern-day hyperloop system. Chen et al. (2012) investigated the effect of different types of head and tail shapes of a maglev train on the pressure ofdraginan evacuated tube. He observed that the train with a linear-shaped head and the blunt tail had the least aerodynamic drag, for given flow conditions. Similarly, in another recent study Yang et al. (2017), on the hyperloop pod shapes, the minimum aerodynamic drag was observed for elliptically shaped head and semicircular shaped tail, for a tube pressure of 100 Pa. There is no general agreement between the shapes that gives the best aerodynamic performance. Furthermore, these conclusions were drawn based on the maximum pressure difference observed across the longitudinal shape of the train or the hyperloop pod. However, this is not a good measure to determine the effectiveness of the shape on the drag. There mightbea possibility of a particular combination of the shape of the head and tail might have less drag but higher pressure difference as compared to a combination that has less pressure difference. It is also found at the energy usage of the pod is not affected significantly by the length ofthe pod. This implies that the system can be scaled to much higher lengths, as compared to the one used in this analysis. This is possible because the friction drag is not scaled significantly (however it increases due to an increase in the length of the pod).In short, because of the effect of the shape on the aerodynamic drag, it is necessary to study the effect ofshapeto reduce the energy consumption and improved the aerodynamic performance. 2.5 CFD Introduction
One of the goals of this thesis is to model the fluid flow around the hyperloop pod using CFD. CFD is a branch of fluid mechanics that uses numerical techniques to simulate and analyze fluid flows based on the conservation laws governing the fluid motion. It isextensively used for parametric studies and flow physics investigations required for various engineering applications. The numerical algorithms solve the governing equations of fluid dynamics viz.
19 continuity, momentum, and energy. These are the mathematical representations of the three fundamental principles.
1. Conservation of mass
2. Newton’s second law, F = ma
3. Conservation of energy The partial differential equation for the first principle, also called the continuity equation is given as, ∂ρ + ▽ · ρu = 0 ∂t Note that all the governing equations are in conservative form, thus signifying that the control volume is fixed in space while fluid moves through it. The momentum equation is derived from Newton’s second law which states that the rate of change of momentum within the fluid element is the summation of the total forces acting on the element.These forces can be divided into two sources: body force (gravitational, magnetic etc.) and surface force (due to shear stress and pressure).The equation is given as,
∂(ρuj) ∂(uiuj) ∂p ∂τij + = + + ρfi ∂t ∂xi ∂xj ∂xi where τij is the shear stress for Newtonian fluid, ( ) ∂ui ∂uj 2 ∂uk τij = µ + − µ δij ∂xj ∂xi 3 ∂xk
The fluid in which the viscous shear stress is directly proportional to the velocity gradient ( ) ∝ ∂V τ ∂t , is called the Newtonian fluid Panton (2013). The behaviour of the fluid can be assumed to be Newtonian, for almost all practical aerodynamic problems Anderson (1995).The third governing principle, conservation of energy Incorporation (2006), states that energy can neither be created nor be destroyed, but it can be converted from one form to another.
∂ ∂ (ρE) + (ρujE + ujp + qj − uiτij) = ρgjuj ∂t ∂xj
20 Since the number of unknowns (density, pressure, three components of velocity and temperature) is greater than the number of equations, we need additional equations to compute the flow field variables. One such equation is Ideal gas law. It is plausible toassume that the gas behaves as ideal gas and follows the general gas equation given as,
p = ρRT
The above mentioned governing set of equation are commonly called as Naiver Stokes Equations and are the cornerstone of every CFD model. 2.6 Turbulence Modeling
Turbulence is a fluid flow phenomenon characterized by irregular, chaotic fluid motion, unsteadiness, rapid mixing, and random change in properties on a wide range of length and time scales. It can be differentiated from the laminar flow based on the Reynolds number. When the Reynolds number is high (∼ 105) the flow is turbulent. In the present work, the Reynolds number in the flow regime around the hyperloop pod is approximately 2 · 105, hence modeling turbulence is essential to capture the flow physics accurately. To observe and investigate the complete fluid motion in a turbulent flow, an ideal approach is to simulate turbulence all over the entire spectrum of spatial and temporal scales ranging from the smallest Kolmogorov scale up to the integral scale. This approach is also known as DNS Wilcox (1998). However, resolving every scale makes DNS computationally expensive. Furthermore, the computational effort required is so high that given the state of the computational power we have today, it is practically impossible to use DNS for almost all applications Wilcox (1998). The most preferred and practical approach to model turbulence for regular engineering applications is to use RANS equations with an appropriate closure model. The RANS equations are time average solution to the Navier-Stokes equations and are derived using a method called Reynolds decomposition. In Reynolds decomposition, the quantity is decomposed into a mean
21 value (time average) and its fluctuating part Wilcox (1998).
” ui =u ˜i + ui
The RANS momentum equation obtained after applying Reynolds decomposition to the NS momentum equations are shown below.
∂ ∂ ∂P ∂ [ ] − − ” ” (ρu˜i) + (ρu˜ju˜i) = + tji ρuj ui (2-1) ∂t ∂xj ∂xi ∂xj
In Equation 2-1, the Reynolds decomposition introduces a new nonlinear term called as Reynolds stress tensor given as,
− ” ” ρτij = ρui uj (2-2)
We now have ten unknowns (six stress terms, three velocity components, and pressure) and only five equations to calculate the flow field variables. This imbalance is known asaclosure problem. One way to solve the closure problem is to model Reynolds stresses using Eddy viscosity model. The six independent terms emanating from the Reynolds stresses, 2-2 are reduced to one scalar field called as Eddy Viscosity. This parameter, νT , can now be estimated using one of the multiple methods which give rise to the so-called different turbulence models. One such method is to use a two-equation model.As the name suggests, this turbulence model uses two equations to calculate the eddy viscosity and there are variety of two equation model which can found at NASA (2019). Because of the accuracy for the flow type modeled in this thesis, we chose the K-omega SST turbulence model Menter (1993); Menter (1994). The K-omega SST is a two equation turbulence model that uses turbulent kinetic energy and the specific turbulence dissipation rate to calculate the eddy viscosity. (*reference). Thismodel is a revised version of the standard K-omega model developed by Wilcox Wilcox (2008). The K-omega SST model combines the standard K-epsilon Launder (2003) and the standard K- omega turbulence model. Because of its superior performance for handling separated flows at an adverse pressure gradient F. R. Menter and Langtry (2012), the K-omega SST model is one
22 of the most widely used turbulence models in the industry for engineering applications. This model is further explained in detailed in the subsequent chapter.
23 CHAPTER 3 METHOD 3.1 Geometry
The primary purpose of the thesis is to study the effect of different shapes of head and tail on aerodynamic drag. Hence, to have effective geometric formulation and transformations so that we can study the effect of the various shapes, the pod shape is parametrized. Itwill also aid in building the surrogate model.Shape parametrization is the process of finding the parametric equation that describes the target geometry. Since we perform analysis on 2D geometry, we describe our geometry using curves. Bezier curves, B-Splines, and NURBS are the most commonly used parametric curves because of their favorable properties [20] [21]. Out of these curves, the Bezier curves are the most efficient to evaluate because we only have a handful of knobs (control points) to tweak the shapes. However, a slight modification in any of the control points defining the Bezier spline modifies the entire global shape ofthe spline, thus making it less effective. B-Splines, on the other hand, require more information to define the curve compared to Bezier curves but provides more control flexibility thanBezier curves. Hence, local shape control is possible without disturbing the whole curve. NURBS is an advanced form of B-Splines and is capable of representing a wide range of complex geometries. Since the geometric shape of the pod involved in our work is not complicated, we use B-Splines which have proven to be helpful in many applications Gray et al. (2018); Yu et al. (2018); Dhert et al. (2017); Garg et al. (2017). A B-spline curve is a linear combination of control points P and B-spline basis functions Hyun et al. (2004), given as,
∑ P (x) = Ni,k(u)Pi (tj−1 ≤ x ≤ tn+1) (3-1)
where, Nij(u) is the blending function given as,
(x − ti)Ki,k(x) (ti+k − x)Ki+1,k−1(x) Ni,j(u) = − ti+k − ti ti+k − ti+1
24 and 1, ti ≤ x ≤ ti+1, Ni,1(u) = 0, otherwise
In the above equation, Pi is the set of control points, ti is the knot value and j is the order of B-spline curve. For non-periodic B-spline curve, the knot values are, 0, 0 ≤ i ≤ k, ti = i − k + 1, k ≤ i ≤ n, n − k + 2, n ≤ i ≤ n + k
In matrix from equation 3-1 becomes, D = NB
B = N −1A, where D is the matrix of data points, N is the matrix of blending functions and B is the matrix of control points. The control points of the B-splines representing the geometry are the design variables. The modifications of these design variables result in different geometric shapes. Hence,they must be chosen meticulously since too many design variables will increase the time and the cost required for analysis, whereas too few variables may not generate a shape that would prove certain potential improvement. An optimum number of design variables is therefore desired to maintain the flexibility within the design space. Based on the literature review, the preliminary design analysis and the time and computational resources available, we have used five and four control points to define the shape of the head and tail, respectively. The geometry is created using CAESES Friendship (2019). The figure 3-1 shows the control points. The head, represented by Curve 1 (blue curve) is constructed using a 4th order (3 degree) B-spline by control points 1,2,3,4,6,and 7. Similarly, the tail and the upper part of the pod is represented by Curve 2 (red curve), and is constructed
25 Figure 3-1. Parametric Pod Geometry using the same order of B-spline, used for head, and control points 7,8,9 and 10. Note that not all the coordinates of the control points are variable. E.g., the length of the pod in our analysis is 2700 mm, hence the x-coordinate of pt.10 is fixed to 2.7(m). Similarly, the pod levitates ata distance of 100 mm above the tube and hence the y-coordinate of points 1,2 and 10 is 0.1(m). The height of the pod is assumed to be 500 mm and hence the y-coordinates of points 6 and 7 become 0.5(m). The table 3-1 shows the values of the coordinates of the control points that are fixed and variable and the table 3-2 shows the constraints imposed on the variables.
Table 3-1. Coordinates of the Control Points of the B-Spline Points x-coordinate (m) y-coordinate (m) 1 variable (k1x) 0.1 2 variable (k2x) 0.1 3 0 variable (k3y) 4 0 variable (k4y) 6 variable (kalpha) 0.6 7 variable (k7x) 0.6 8 variable (k8x) variable (k8y) 9 variable (k9x) variable (k9y) 10 2.7 0.1
3.2 Mesh
The design and construction of high-quality mesh are crucial to the success of any CFD analysis. A poor quality grid can severely affect the numerical stability of the solver and the accuracy of the solution. An ideal case would be to have a high-resolution grid, but the downside is that it demands higher computational effort and increases CPU run times.
26 Table 3-2. Constraint of the variables of the control points Variable Minimum Maximum k1x 0 0.5 k2x 0 1 k3y 0 0.9 k4y 0.2 0.35 kalpha 10 60 k7x 0.5 1 k8x 2 2.5 k8y 0.5 0.6 k9x 0.5 1 k9y 0.5 1
Therefore, an optimal balance needs to be achieved between the desired level of finesse and the computational cost. There are two types of grid, structured grid, and unstructured grid. The structured grid is characterized by regular connectivity of elements, while in an unstructured grid, the elements are connected irregularly. Both the grids have their pros and cons. The structured grid requires less storage memory, whereas the unstructured grid requires more memory. A well designed structured grid leads to faster convergence and accurate solution, which is not always the case with the unstructured grid. Creating the structure grid on a complex geometry can take up to several months. On the other hand, the unstructured grid offers high flexibility and automation while dealing with complex geometries. The geometry of the pod involved in the simulation is relatively simple and has no complicated features. Mainly, for this reason, we use a structured grid in our work. The structured grid is created using ICEM CFD(ref). Since the shape of the head and the tail changes during analysis and during surrogate modeling, we have written a script to automate the meshing process and to have a high-quality grid for each simulation. To accurately predict the flow behavior in the boundary layer where a large velocity gradient exists, we haveadded the prism layers. Since we use the K-omega SST turbulence model, we add the prism layers in such a manner that y+ ≤ 1 everywhere on the pod geometry. The Figure 3-2 shows the structured grid based on the level of finesse created on a semicircle nose and a blunt tail.
27 A
B
C
Figure 3-2. The Structured Grid around Pod with Semicircle head and Blunt tail. A) Coarse Mesh(0.4 million cells); B) Fine Mesh (0.6 million cells); C) Extra Fine Mesh (0.9 million cells).
3.3 Numerical Solver
3.3.1 Finite Volume Method
FVM is a numerical technique of discretizing the partial differential equation representing the conservation law over differential volumes. Almost all the commercial solvers, including ANSYS Fluent, is based on the FVM. In this method, the flow domain is divided into finite-size non-overlapping sub-domains called as control volumes. The governing differential equations are then integrated over each discrete control volume and transformed into an algebraic equation. This system of an algebraic equation is then solved numerically to compute the
28 value of a dependent variable for each of the subdomain. The integration of the governing differential equation over each control volume ensures strict conservation in each celland globally in the domain. Because of this inherent conservation property, the FVM is the most preferred in CFD as compared to the finite difference and finite element method. Furthermore, the formulation of the FVM on the unstructured grid makes it more robust and fast. 3.3.2 Pressure Based Solver and Density Based Solver
The governing fluid dynamics equations can be solved in ANSYS fluent usingtwo different solvers, the PBS and the DBS. Historically speaking the PBS was designed andhas been traditionally used for incompressible flows while the DBS was designed to be used for compressible flows. In the recent Fluent version, both the solvers have been redesigned and formulated in such a way that they can be extensively used for a broad category of fluid flow conditions. To obtain the velocity field, both these solvers utilize momentum equations. The DBS is preferred for complex and highly compressible flows where the coupling of the energy equation is essential. This coupling makes the density-based solver more memory extensive since it has to solve coupled matrix equations in one go. In our present work, the fluid flow is mildly compressible, and there is no complex flow interaction involved. Hence, we use PBS over DBS. As discussed earlier, we solve the momentum equations for the velocity field. However, to solve the momentum equations, we need pressure field distribution since the pressure gradient term appears in the equation. In short, the velocity can be obtained from the momentum equations only if the pressure is known. However, calculating pressure is not straightforward since the continuity equation cannot be used to obtain the pressure field. ANSYS Fluent handles this pressure velocity coupling using two different algorithms named as segregated solver and a coupled solver. The segregated solver solves the momentum equations for an unknown velocity using guessed pressure. The obtained velocity is then used to see if the continuity equation is satisfied or not. If the resulting velocity does not satisfy the continuity equation, the pressure
29 correction equation is solved, and the pressure field is updated. With this updated pressure field the velocity field is also updated. This process is repeated until the continuity, andthe momentum of equations are satisfied. Since only one scalar equation is solved at a time,the name segregated solver. Moreover, only one discrete equation needs to be stored at a time, which results in a lower memory requirement. However, because of the iterative nature of the solver, the convergence is often slower. The coupled solver, on the contrary, solves for the velocity and the pressure in a coupled fashion and hence the name coupled solver. Since all the discrete equations need to be stored at a time, the memory requirement is high as compared to the segregated solver. However, the coupled solver provides an accelerated convergence. Furthermore, the coupling makes this algorithm more stiff and less unstable. As discussed in the section 3.4, this solver satisfies our stability requirements and provides accelerated convergence, we use pressure-based coupled solver for our analysis. 3.4 Numerical Model
When studying the effect of different shapes of head and tail, the pod is assumedto be traveling at a speed of 103m/s, and the pressure inside the tube is maintained that 860 Pa. The blockage ratio, in this case, varies from 0.25 - 0.6. While investigating the effect of pressure, the pod is assumed to travel at a speed of 103 m/s maintaining a constant blockage ratio of 0.35 and the pressure varying from 10 Pa - 100 KPa. Similarly, while studying the effect of velocity, the pressure inside the tube is maintained at 860 pascals, and theblockage ratio of 0.35 is kept constant. The velocity, in this case, varies from 50 m/s to 300 m/s. The flow being mildly compressible, in most of the cases, the thermal dependence of viscosity is taken into account using Sutherlands law. We also assume that the static temperature inside the tube is 300 K, and the air follows ideal gas law, i.e., the density depends on the ambient pressure inside the tube. The flow filed is also assumed to be steady and viscous. The
30 Reynolds number for the flow field is given as,
ρV L R = = 151790.71 e µ
Note that the characteristic length is assumed to be represented by the length of the pod. Since the Reynolds number is of the order of 105 the flow field will be turbulent. Hence an appropriate turbulence model needs to be chosen. We have therefore used K-omega SST model as discussed in the section 2.6. Any simulation around the Kantrowitz limit is unsteady because of the chocking Opgenoord and Caplan (2018). This unsteadiness prevents the steady solution from converging. However an excellent stiff solver and careful set up of the inletand outlet boundary conditions will help to get around with this problem. The boundary conditions used are described in detail below. 3.5 Boundary Conditions
A well-posed CFD problem requires relevant information of the flow variables to be specified at the boundary of the flow domain known as boundary conditions. These boundary conditions serve as a constraint and establish the uniqueness of the flow field for the stated problem. To define these boundary conditions, one needs to identify the number ofthe boundary conditions required, the location of the boundaries, and the form of the boundary conditions. This depends on the physics we are interested in and the physical model that is employed for the problem. The boundary conditions also have direct repercussions on the convergence and stability of the solution, and hence, must be chosen prudently. It is a common practice to use pressure inlet boundary condition at the inlet for compressible flows. This ensures that the total pressure remains fixed while themassflux can vary Inc. (2006). However, in our current work, the possibility of the choked flow around the pod will cause pressure to build up in front of the pod. In that case, it is necessary to ensure that the prescribed mass flow rate is matched. Therefore, we use mass flow boundary condition at the inlet. Specifying the mass flux gives flexibility for the total pressure tovary
31 and adjust according to the interior solution. At the outlet, a pressure far-field boundary condition is used to model, free stream conditions at infinity where free stream Mach number and static conditions are specified. We use wall boundaries to bound the solid and fluid regions. Due to the viscosity ofthe fluid, the no-slip boundary condition is imposed at the wall. When the fluid comes incontact with the surface of the pod, the fluid will not have any velocity relative to the pod. Hence the fluid in direct contact with the pod will stick to the surface of the pod and no-slipwillbe observed. It means that the tangential velocity of the fluid equals the velocity of the wall and the normal component of the velocity is zero. For simulations involving the flow around a vehicle like a car or an airplane, it is ageneral practice to assume that the vehicle is stationary while the walls are moving at the velocity at which the vehicle is expected to travel. We, therefore, assume that the pod is stationary and apply a tangential velocity of 103 m/s at the wall. The inlet of the tube is located at a distance of one pod length since no complex fluid flow interaction is expected in that region. The outlet, on the other hand, is placed at a distance of four pod length, form the rear of the pod, so that it doesn’t influence the upstream flow. 3.6 K-omega SST Turbulence Model
The approximate Reynolds number calculated for our prototype (3.4) suggests that the pod will experience turbulent flow. This necessitates the use of an appropriate turbulence model to capture the physics associated with the flow. As mentioned earlier we have used the K-omega SST turbulence model because of its ability to handle the adverse pressure gradients well. The SST K-omega model is a hybrid model that effectively blends the robust and accurate formulation of standard K-omega model in the rear wall region with free stream independence of the K-epsilon model in the far-field Galindo et al. (2013). In the SST K-omega, the
32 transport equation for k Wang et al. (2016) is given by, ( ) ∂ ∂ ∂ ∂k (ρk) + (ρkui) = Γk + Gk − Yk + Sk (3-2) ∂t ∂xi ∂xj ∂xj while the transport equation for ω Wang et al. (2016)is given as, ( ) ∂ ∂ ∂ ∂ω (ρω) + (ρωuj) = Γω + Gω − Yω + Dω + Sω (3-3) ∂t ∂xj ∂xj ∂xj
The first transient term, in Equation 3-2 and 3-3, is the rate of change of k and ω, respectively. The second term represents the transportation of k and ω by convection. The third term is the diffusive transport of k and ω due to turbulent diffusion. Gk represents the generation of turbulent kinetic energy due to the mean velocity gradient. Similarly, Gw represents the generation of ω. Yk and Yg represent the dissipation of k and ω due to turbulence.The term
Dw in the Equation 3-3 is called a cross-diffusion term.It is this term that is responsible for the transition of the modeling from epsilon to omega thereby marrying standard K-epsilon and the standard K-omega models, by using a blend function based on the wall distance. In other words, by using wall distance as a switch, SST works like K-epsilon in far filed and K-omega near the target geometry. The cross diffusion term is given as,
1 ∂k ∂ω Dω = 2(1 − F1)ρ (3-4) ωσω,2 ∂xj ∂xj
Since SST K-omega model combines the best of two models, its ability to switch between the models makes it very popular for external aerodynamics application. Within the inner parts of the boundary layer the blending function in the Equation 3-4, F 1 = 1. As a result, the cross-diffusion term disappears from the Equation 3-3. The equation now resembles the equation in the standard K-omega model. This switches the SST K-omega formulation to the standard K-omega model, allowing us to use the model directly through the viscous sub-layer without any need of the damping function. Similarly, in the free stream the blending function F 1 = 0, thus switching the SST K-omega to the K-epsilon model and eliminating the problem of sensitivity associated with the free stream conditions.
33 3.7 Convergence and Verification
Verification and Validation are the cornerstones of any simulation study. It is the primary method by which we can build confidence and quantify the obtained CFD results. Hyperloop is a fairly new concept and an unconventional system. And everything has to be designed and tested from scratch. As of today, no experimental aerodynamic data is available for the hyperloop pod. Hence the convergence of the obtained CFD simulation is the only way we can verify the accuracy and reliability of our results. Verification also involves identifying the errors and quantifying them. The errors are broadly classified into three main categories.
Figure 3-3. Effect of Head shape on the drag for different operating pressures and velocity
1. Discretization Error: It is the difference between the exact solution of the governing flow equations and the solution obtained by representing those equations as algebraic expressions in the discrete domain of space and time
2. Iterative Convergence Error: Almost all the numerical solutions are obtained using the iterative method. Hence we need a stopping criterion to stop this process at a
34 certain level (to reduce numerical effort). The difference between the exact solution and this iterative solution contributes to the iterative convergence error.
3. Physical Modeling Error: This error can be attributed to the difference between the real world physical problem and the solution to the mathematical model representing it. Discretization error depends on the quality of the grid, which implies that the accuracy of the solution is grid-dependent. To ensure that the solution is grid-independent, we perform a systematic exercise called a grid convergence study. During this study, a series of grids ranging from coarser mesh to finer mesh is created. The variation of the figure of merit, i.e. drag(in our case) is then analyzed for each grid. The Figure 3-3 shows the variation of the drag for the grid levels shown in Figure 3-2. With the increase in the grid’s finesse, the drag becomes saturated, and we can see no further change. This is because, with the increase in the finesse, the spatial discretization error asymptotically approaches zero, thereby confirming that the solution is independent of the grid. The Figure 3-3 shows the time required to obtain the solution at each grid level. When the size of the grid changes from 0.6 million cells to 0.9 million cells, only a 0.08% difference in the drag is observed. However, the time required for the computation is comparatively high. Since the difference in the drag’s value is low, and considering the time and the computational resource available, we use approximately 0.6 million cells for each CFD analysis. Residual Analysis: The residual is a local imbalance of a conserved variable within each control volume for each algebraic equation solved. Hence, each cell of the computational domain will have a residual value. Since the residual value directly quantifies the error in the solution ofthe system of equations, analyzing them will help to determine iterative solutions convergence. For these reasons, analyzing residuals is one of the most fundamental measures while determining the convergence of a solution. However, because of the iterative nature of the solution, the residual values will never be precisely zero. The lower the value of the residual, the higher the accuracy of the solution. The Root Mean Square residual level of 1e-4 is considered to be loosely converged; the level of 1e-5 is considered to be well converged while the level of 1e-6 is
35 Figure 3-4. Residuals
Figure 3-5. Net Mass Flow Rate (across Inlet and Outlet) Vs Iterations considered to be tightly converged Putra Adnan and Hartono (2018). The figure 3-4 shows the residuals observed, for a CFD analysis of hyperloop pod with semicircle head and blunt tail. Another way to verify the accuracy of our final solution is to ensure if mass, momentum, and energy are conserved over the whole domain. This is because the CFD analysis involves
36 Figure 3-6. Plot of Drag Coefficient Vs Iterations
Figure 3-7. Plot of Drag Vs Iterations
solving the governing fluid equations viz. conservation of mass, conservation of momentum, and conservation of energy. The figure 3-5 shows the net mass flow rate observed across the inlet and outlet of the tube, confirming that the mass flow rate is conserved. (Note: Thesmall
37 imbalance in the mass is due to the numerical representation of the physical system arising because of the iterative convergence error and hence will ever be precisely zero) Another way to verify the accuracy of the result is to monitor the convergence of integrated quantities like a drag, drag coefficient, etc. The steady-state analysis is deemed to be converged if the solution field does not change for every iteration. Hence, the convergence of the terms mentioned above will directly imply that the solution is converged. The Figure 3-7 and 3-6 is the plot of drag and drag coefficient for iterations and we can conclude that, the solution is converged.
38 CHAPTER 4 SURROGATE MODELING We have created a parametric geometry to study the effect of different shapes of head and tail on the aerodynamic drag. As discussed earlier, ten design variables govern the shape of the head and the tail. Each design parameter has an impact on the drag of the vehicle; however, the extent of the impact is unknown. This can be determined by calculating the gradients of each design variable with respect to the other known as sensitivity. Sensitivity can be calculated in numerous ways. Since the number of design variables is large, the most commonly used method to compute the sensitivity is to use a surrogate model. The response surface models or surrogate models are compact, scalable analytical models that approximate the multivariable input-output behavior of a complex system. It consists of constructing a mathematical model of the quantity of interest called a surrogate or a response surface from a limited number of observations. In short, a surrogate model is an inexpensive approximate model representing the expensive high fidelity experiment or observation. The gradients then can be easily computed using the surrogate model. Moreover, a surrogate model gives a better picture and detailed understanding of the design space. There are many different methods available to build the surrogate such as polynomial response surface Queipo et al. (2005), Kriging Forrester and Keane (2009), radial basis function Wild et al. (2008), etc. to name a few. Below mentioned are the steps performed to create a surrogate model:
1. It consists of sampling the design space to obtain the design points. The design of experiment methods is used to locate the position of the design points. The doe methods will be explained in the subsequent sections.
2. The necessary information is then gathered at the design points by conducting the experiments (simulations in our case).
3. A mathematical model is constructed using gathered information called as surrogate.
4. The obtained surrogate is then validated by using the verification points. This is done by comparing the values obtained by high-fidelity simulation against those obtained by the surrogate model at the verification points. If the quality of the surrogate isnotas desired, the surrogate model is refined using the refinement points.
39 5. Finally, the gradients are calculated from the obtained surrogate When we fit a surrogate, the set of points where we sample the data is called asampling plan or design of experiment. When, the data is very noisy so that noise is the main reason for errors in the surrogate, sampling plans for linear or quadratic regression like Central composite design are most appropriate. This DOE tends to favor points on the boundary of the domain. On the other hand, when the error in the surrogate is due to the unknown shape of the actual function and is not subjected to noise sampling plans for Kriging like LHS or OSF are preferred. These DOE’s are often called as space-filling DOE since the design points are evenly spaced in the domain. LHS is one of the most widely used DOE Beachkofski and Grandhi (2002), Giunta et al. (2003). It is an advanced form of Monte Carlo sampling in which no design point shares the row or column of the design space with any other point. Another type of space-filling DOE is OSF. It is an optimized form of LHS, in which the design points are uniformly distributed across the design space. The uniformity is obtained by maximizing the distance between the design points, ensuring full coverage of the design space. The LHS method can sometimes result in the closed, uneven grouping of the design points and so can skip some parts of the design space. Due to the maximization of the distance between the points and more uniform distribution of points, OSF provides better coverage of the design space and can address the extremes more effectively. Since the computer simulation is not entirely subjected to noiseand with the added benefit of uniform distribution of design point, we use OSF DOE in our present work. To ensure the accuracy and have confidence in the obtained results, we use two different models to build the surrogate. We have therefore, used the polynomial regression model (polynomial response surface) and Kriging to build our surrogate. 4.1 Polynomial Regression Model
The PRG is a polynomial approximation of the function of interest f and Np basis functions zj based on the Ns sample values of the function of interest. The sampled data is
40 fitted using a least square regression technique. The mathematical formis,
∑Np 2 fi(z) = βjzj + ϵi,E(ϵi) = 0,V (ϵi) = σ (4-1) j=1
where ϵ is a random error, considered to be independent and normally distributed with mean 0
2 and variance of σ , βj are the unknown coefficients, and zj is the set of basis functions Draper and Smith (1998). The Equation 4-1, can be written in matrix form as,
f = Xβ + ϵ, E(ϵ) = 0,V (ϵ) = σ2
where X is a Ns × Np matrix of basis functions with the design variables for sampled points and β is the vector of the regression parameters, which can be calculated as,
βˆ = (XTX)−1XTf
We have used a second order polynomial (No = 2), that can be expressed as,
∑No ∑No ∑No ˆ f(x) = β0 + βixi + βijxixj i=1 i=1 j≤1 with the basis functions being the monomials 1, xi, and xj. The second order PRG provides a best compromise between the modeling accuracy and computational expense as compared to the other polynomial methods [64].One of the main advantages of this method is its ability to smooth out the noise and capture the global trend variation. Hence this model is commonly employed when there is a noise associated with the data. 4.2 Kriging Regression Model
Kriging is a Gaussian process-based interpolating method. The mathematical model of Kriging Journel and Huijbregts (1976); Simpson et al. (2001) assumes the function of interest as a linear combination of global trend function g(x)T α and a Gaussian random function Z(x) given as, f(x) = g(x)T β + Z(x)
41 T T where g(x) = [g1(x) g2(x)... gk(x)] are known functions, α = [β1 β2... βk] are the unknown model model parameters, Z(x) is a realization of normally distributed Gaussian random process with zero mean and variance σ2. The regression part m(x)T β globally approximates the function while Z(x) addresses the localized variations. The covariance matrix of Z(x) is given as, [ ( ) ( )] ([ ( )]) Cov Z x(i) ,Z x(j) = σ2R R x(i), x(j)
(i) (j) where R is a p × p correlation matrix with Rij = R(x , x ), which is a correlation function between sampled data points x(i) and x(j). The most general and popular choice is the Gaussian correlation function [ ] ∑n R(x, y) = exp − θ |x − y |2 k=1 k k k
Kriging interpolation is one of the most popular and widely used methods to build a surrogate for engineering design problems, which are nonlinear, multivariable, multimodal, and most importantly, noise-free Journel and Huijbregts (1976); Simpson et al. (2001); Kleijnen (2009); O’Hagan (1978). Since the computer simulation is not quite subjected to noise, and it is compact and cheap to evaluate Rasmussen (2004), we use Kriging to build the surrogate. 4.3 Surrogate Results
Based on the number of design variables, the Design Exploration tool within ANSYS automatically calculates, the number of the sample points, which in this case turns out 240 samples. To obtain good, accurate, and reliable results, rule of thumb commonly followed is to double the number of design points, if the resources permit. Hence we add 120 samples to the previous samples. As discussed in the section, we validate our model using 60 verification points. The Figure 4-1 shows the plot of drag values obtained from the polynomial and Kriging regression model vs actual observed values. The goodness of fit is used to measure the accuracy of the surrogate models and is shown in the Table 4-1. The desired value of the COD is 1, while that of root mean square value is 0. In the case of the polynomial regression model, COD is 0.99, whereas the root mean square value is 0.101
42 A
B
Figure 4-1. Plot of Drag obtained from the Response Surface vs Actual Observed Values. A) PRG; B) Kriging Regression Model.
Table 4-1. Goodness of Fit for Polynomial and Kriging Regression Model Goodness of Fit PRG Kriging Expected Value Coefficient of 0.99396 1 1 Determination Root Mean Square 0.10137 5.90e−8 0 Relative Absolute 3.4 2.9 0 Maximum Error(%)
43 for sampled points and 0.07 for verification points. It implies that the model is sound and accurate enough to represent the actual function. The COD of is not a reliable measure of fit for Kriging because of its inherent ability to pass through the data points. We therefore rely on the root mean square value of the verification points which is 0.0659. Based on the Goodness of the fit we conclude that the Kriging is superior to the Polynomial regression model inour case. Still we use both the models to build the surrogate and compute the sensitivities so that we can have confidence in our results.
44 CHAPTER 5 RESULTS 5.1 Effect of Shape of Head
In this study, we analyzed how different shapes of head affect the drag under varying blockage ratios. We perform the analysis on three distinct nose shapes viz. semicircle nose, arched shaped nose, and splined elliptic nose. We have used a rectangular tail (blunt tail) which remains unchanged for each head shape, and the blockage ratio varies from 0.25 - 0.6. The pod travels at a speed of M = 0.3, and the pressure inside the tube is 860 Pa. The Figure 5-1 shows the effect of head shape on the drag of the pod for various blockage ratios.
Figure 5-1. Effect of Head shapes on drag for different Blockage ratio
First, as the blockage ratio increases, the drag increases. This is obvious because an increase in the blockage ratio decreases the area available for the flow between the pod and the tube, thereby speeding up the fluid in the throat area. This increase in the fluid velocity directly increases the drag. Second, despite the shape of the head, the aerodynamic drag remains approximately the same for blockage ratio ranging from 0.25 - 0.55. This signifies that the shape of the head has no profound impact on the aerodynamic drag regardless of
45 the blockage ratio which can be attributed to the lower internal tube pressure. However, at a blockage ratio of around 0.6 we start to see small effect because of the chocking of the flow and formation of shock waves.
Figure 5-2. Effect of Head shapes on drag for different operating Pressures
The Figure 5-2 shows the effect of head shapes on the aerodynamic drag under varying tube pressures. It can be seen that when the pressure inside the tube ranges between 10 Pa - 1 KPa, the pod with semicircle nose, arched nose and splined elliptic nose experience almost the same amount of drag. However, when this tube pressure exceeds the value of 1000 Pa, a gradual change is observed in the drag experienced by the pod with all three head shapes which later becomes significant after 10 KPa. Since the change in the blockage ratio doesnot alter the pressure inside the tube and the pressure inside the tube is below 1 KPa (860<1000) we see no obvious effect of head shape on the aerodynamic drag for varying blockage ratios. As mentioned earlier, for a hyperloop pod traveling through an evacuated tube, the pressure drag dominates the viscous drag. This pressure drag emanates because of the pressure differential caused by the high-pressure region in front of the pod and the low-pressure wake
46 Figure 5-3. Distribution of Pressure Coefficient around the nose for different Head shapes
region at the rear of the pod. Here, since the shape of the tail remains the same, the area of the low-pressure turbulent wake region formed at the rear of the pod also remains the same. As a result,the pressure drag entirely depends on the shape of the head. In other words, it depends on the pressure distribution around the head and importantly on the maximum pressure experienced by the head. We express this pressure distribution around the body in terms of a dimensionless number called a Pressure Coefficient. The Figure 5-3 shows the
distribution of Cp around the nose of the pod with different shapes of head. Since the shape of the head is different, the distribution of Cp is also different, which is obvious as seen from the graph. However, one important thing to note is that the maximum value of the Cp observed for all the shapes of the head is almost same, and hence the drag experienced by the pod with different head shapes is approximately same. This only holds as long as the pressure
47 Figure 5-4. Pressure Distribution across the Semicircle shaped head inside the tube is below 1000 Pa. As discussed earlier, when the pressure exceeds 1 KPa, a significant change is observed in the values of Cp for different head shapes. Now, for the pressures below 1 KPa, even though the maximum value of Cp is almost the same, there is still room for improvement. For e.g. the semicircle nose has the highest value of Cp (maximum) of
2.88, while the splined elliptic shaped nose has the minimum value of Cp (maximum) of 2.85 when the blockage ratio is 0.35. The semicircle nose has a higher value of Cp because of the larger area of high pressure, as seen in the figure 5-4. Comparing Figures 5-4, 5-5, and 5-6 we see that the elliptic nose has the least area of high pressure of all three head shapes and hence has 1.2% and 0.73% less drag than a semicircle and arched shaped head, respectively. The This signifies that while optimization of the shape of the head, minimizing the area ofhigher pressure can be a starting point. For a specific tube pressure, there is an absolute velocity called as a limiting velocity, below which the head shape has no pronounced effect on the aerodynamic drag of the vehicle.
48 Figure 5-5. Pressure Distribution across the Arched shaped head
E.g., when the pressure in the tube is 860 Pa, no effect of head shape is observed on thedrag for velocities up to 100 m/s and when this pressure reduces to 100 Pa, no effect is noticed for velocities up to 200 m/s as seen in Figure 5-7. This is because, for a given tail and head shape, the pressure drag is a function of the maximum pressure experienced by the nose of the pod. This maximum total pressure is an algebraic sum of static and dynamic pressure. When the pod travels at a relatively lower velocity (< 200 m/s), the dynamic pressure is low. It is the static pressure in the tube that drives the total pressure, which determines the limiting value of the velocity. As discussed earlier, when this value of the total pressure exceeds the critical value, the shape of the head shows its impact on the aerodynamic drag. When the pressure inside the tube is high, there is less room left for the dynamic pressure until the value of the total pressure reaches the critical value. As a result, the effect of the head shape kicks in at a lower velocity and vice versa. When the pod travels at a very high velocity, depending upon the blockage ratio, there is a
49 Figure 5-6. Pressure Distribution across the Splined Elliptic shaped head
possibility of chocking of the flow as well as the formation of the shock. Hence, regardless of the static pressure in the tube, the total pressure will be high. If the total pressure exceeds the value of the critical pressure, the shape of the head will affect the aerodynamic drag. 5.2 Effect of Shape of Tail
Similar to the study of head shape, we analyzed the effect of different shapes of the tail on the aerodynamic drag under varying blockage ratios. We perform the analysis on three distinct tail shapes viz. rectangular tail (blunt tail), elliptic tail, and splined tail. We have used an arched shaped nose that remains unchanged for each tail shape, and the blockage ratio varies from 0.25 - 0.6. The pod travels at M = 0.3, and the pressure inside the tube is 860 Pa.The Figure 5-8 shows the effect of tail shape on the drag of the pod for various blockage ratios.
50 Figure 5-7. Effect of Head shape on the drag for different operating pressures and velocity
Figure 5-8. Effect of Tail shapes on drag for different Blockage ratio
51 Figure 5-9. Maximum Pressure generated on the head due to different shapes of Tail
First,the aerodynamic drag increases as the blockage ratio increases. This effect of the blockage ratio is like that observed with the head shape. As opposed to the shape of the head, the shape of the tail has a significant impact on the aerodynamic drag for a given blockage ratio. E.g., when the blockage ratio is 0.25, the splined shaped tail has 40% less drag as compared to the elliptic tail and 60% less drag compared to rectangular tail. Furthermore, as the blockage ratio, the effect of the shape of the tail becomes even more significant. Thetail shape also affects the maximum pressure generated on the head as shown in Figure 5-9.
Table 5-1. Maximum Pressure Generated on Head for different Tail shapes Tail Shape Maximum Pressure (Pa) Blunt 1015.26 Elliptic 1009.21 Splined Shape 1004.30
Here, for a given blockage ratio, as the shape of the head remains unchanged, it is the shape of the tail that ultimately decides the pressure drag. With a rectangular tail, the turbulent wake region, at the rear of the pod, has a more substantial area as compared to the
52 Figure 5-10. Velocity Distribution across the pod with Blunt Tail other two shapes of the tail as shown in Figures 5-10, 5-11, and 5-12. As a result, the pressure differential is significant and hence, the aerodynamic drag.
Figure 5-11. Velocity Distribution across the pod with Elliptic Tail
Figure 5-13 is the plot of shear stress distribution along the top surface of the hyperloop pod. The figure 5-13 shows that the shear stress becomes negative for the pod with an elliptic and splined shaped tail, implying the separation of the boundary layer. However, the shear stress is more negative for an elliptically shaped tail as compared to the splined shaped tail.
53 This implies that the adverse pressure gradient is higher for an elliptically shaped tail because of the sudden change in the tail’s curvature.
Figure 5-12. Velocity Distribution across the pod with Splined shaped Tail
Figure 5-13. Effect of Head shape on the drag for different operating pressures and velocity
54 Moreover, due to the sudden change in the curvature, the value of the shear stress changes from positive to negative over a short span on the surface of the pod as opposed to the gradual change in case of the splined shaped tail. As a result, the elliptically shaped tail has a higher drag compared to the splined shaped tail.
Figure 5-14. Effect of Tail shape on Drag for different operating Pressures
Like Blockage ratio, the effect of the tail shape becomes even prominent with increase in pressure and velocity, and can be seen in Figure 5-14 and 5-15. However, for any increase in velocity beyond 250 m/s the shape of the tail has no effect. This is because the flow gets chocked and no information regrading the shape can be transmitted back in the flow. Hence regardless of the blockage ratio, pressure and velocity, the tail shape significantly affects the drag experienced by pod. 5.3 Sensitivity Analysis
The Figure 5-16 and Figure 5-17 the global sensitivities of the design variables obtained using the polynomial and Kriging regression model, respectively. The sensitivities obtained from both the surrogate models are approximately the same. From both the figures, we can see that the drag is highly sensitive to the design variable k8x, k8y, and k9y. However, the effect of k9x is not as pronounced as seen from Polynomial Regression Model compared to the Kriging Regression Model. These variables, govern the
55 Figure 5-15. Effect of Tail shape on Drag for different Velocity
Figure 5-16. Sensitivity based on Polynomial Regression Model
56 Figure 5-17. Sensitivity based on Kriging Regression Model shape of the tail, confirming that, the shape of the tail has a significant impact onthedrag. The design variable k7x and kalpha control the curvature of the top surface of the head. This is the point where the flow has a maximum velocity during travel. If the curvature ishighthe velocity will be too high, and it may exceed the speed of sound thus leading to the formation of shocks and hence we see its effect on the drag. We also see that the design variable k1xand k4y also effect the drag but not significantly. These parameters controls the amount ofarea where the flow becomes stagnant. As discussed earlier, higher the area at the stagnation point, higher is the drag and hence the effect of the variable. In short, all the deign variables, affect drag but but those governing the shape of the tail affect significantly.
57 CHAPTER 6 CONCLUSION This work is aimed at studying the effect of different shapes of head and tail onthe aerodynamic drag for varying blockage ratios, pressure, and velocity. It is found that the shape of the head has no profound impact on the aerodynamic drag irrespective of the blockage ratio. However, if the blockage ratio becomes high (∼ 0.6), the chocked flow and the shock waves set in affecting the aerodynamic drag. Moreover, for a particular value ofthetube pressure, there is limiting velocity, below which the shape of the head will not affect the aerodynamic drag significantly. Lower the value of the pressure, higher is the limiting velocity and vice versa. Since the shape of the head had no prominent effect, fewer resources can be spent while optimizing the shape. One such example can be to use less number of design variables while defining the shape of the head. Despite less effect, there is still a littleroomfor improvement. While designing the shape, care must be taken to keep the least possible area of maximum pressure (stagnation point), which will aid in reducing the aerodynamic drag. On the other hand, the shape of the tail is found to have a significant impact on the aerodynamic drag regardless of the blockage ratio, pressure, and velocity. This effect becomes even more prominent with the increase in the Blockage ratio, pressure, and velocity. However once the flow gets chocked, no effect of the tail shape will be seen on the aerodynamic drag. The maximum pressure generated on the head also depends on the shape of the tail. Because of its dominance, the design of the tail becomes of paramount importance to improve the aerodynamic performance of the vehicle and hence more resources should be allocated, and utmost care should be taken while designing the shape of the tail.
58 APPENDIX ICEM CFD SCRIPT
# Replay file ICEM CFD 19.1 in Workbench 2.0 Framework # unloading mesh ic_unload_mesh # unloading blocking ic_hex_unload_blocking # delete empty parts ic_delete_empty_parts ic_geo_set_part curve EDGEE5505 INLET 0 ic_delete_empty_parts ic_geo_set_part curve EDGEE5605 TWALL 0 ic_delete_empty_parts ic_geo_set_part curve EDGEE5705 OUTLET 0 ic_delete_empty_parts ic_geo_set_part curve EDGEE5405 BWALL 0 ic_delete_empty_parts ic_geo_set_part curve EDGEE5805 NWALL 0 ic_delete_empty_parts ic_geo_set_part curve EDGEE6005 PTWALL 0 ic_delete_empty_parts ic_geo_set_part curve EDGEE5905 PBWALL 0 ic_delete_empty_parts ic_geo_new_family V_FLUID ic_boco_set_part_color V_FLUID ic_hex_unload_blocking ic_hex_initialize_mesh 2d new_numbering new_blocking V_FLUID
59 ic_hex_unblank_blocks ic_hex_multi_grid_level 0 ic_hex_projection_limit 0 ic_hex_default_bunching_law default 2.0 ic_hex_floating_grid off ic_hex_transfinite_degree 1 ic_hex_unstruct_face_type one_tri ic_hex_set_unstruct_face_method uniform_quad ic_hex_set_n_tetra_smoothing_steps 20 ic_hex_error_messages off_minor ic_hex_find_comp_curve EDGEE5605 ic_hex_set_edge_projection 13 21 0 1 EDGEE5605 ic_hex_find_comp_curve EDGEE5705 ic_hex_set_edge_projection 19 21 0 1 EDGEE5705 ic_hex_find_comp_curve EDGEE5405 ic_hex_set_edge_projection 11 19 0 1 EDGEE5405 ic_hex_find_comp_curve EDGEE5505 ic_hex_set_edge_projection 11 13 0 1 EDGEE5505 ic_set_global geo_cad 0 toptol_userset ic_set_global geo_cad 0.008 toler ic_geo_new_family GEOM ic_boco_set_part_color GEOM ic_delete_geometry point names {pnt.00 pnt.01 pnt.02 pnt.03 pnt.04 pnt.05} ,→ 0 1 ic_delete_geometry point names {pnt.06 pnt.07 pnt.08 pnt.09} 0 1 ic_delete_geometry point names {pnt.10 pnt.11 pnt.12} 0 1 ic_point {} PART_1_1_1 pnt.00 VERT15+vector(2.4,0,0)
60 ic_point {} PART_1_1_1 pnt.01 VERT15+vector(5.7,0,0) ic_point {} PART_1_1_1 pnt.02 VERT14+vector(0,0.75,0) ic_point crv_par PTWALL pnt.03 {EDGEE6005 0.5} ic_point crv_par PTWALL pnt.04 {EDGEE6005 0.75} ic_point crv_par NWALL pnt.05 {EDGEE5805 0.4} ic_point crv_par NWALL pnt.06 {EDGEE5805 0.75} ic_hex_unstruct_face_type ic_hex_set_unstruct_face_method ic_hex_split_grid 13 21 pnt.00 m PART_1_1_1 INLET TWALL OUTLET BWALL NWALL ,→ PTWALL PBWALL V_FLUID GEOM VORFN ic_hex_split_grid 34 21 pnt.01 m PART_1_1_1 INLET TWALL OUTLET BWALL NWALL ,→ PTWALL PBWALL V_FLUID GEOM VORFN ic_hex_split_grid 11 13 pnt.02 m PART_1_1_1 INLET TWALL OUTLET BWALL NWALL ,→ PTWALL PBWALL V_FLUID GEOM VORFN ic_hex_split_grid 11 41 VERT17 m PART_1_1_1 INLET TWALL OUTLET BWALL NWALL ,→ PTWALL PBWALL V_FLUID GEOM VORFN ic_hex_split_grid 34 38 VERT18 m PART_1_1_1 INLET TWALL OUTLET BWALL NWALL ,→ PTWALL PBWALL V_FLUID GEOM VORFN ic_hex_split_grid 56 38 pnt.03 m PART_1_1_1 INLET TWALL OUTLET BWALL NWALL ,→ PTWALL PBWALL V_FLUID GEOM VORFN ic_hex_mark_blocks unmark ic_hex_mark_blocks superblock 22 ic_hex_mark_blocks superblock 10 ic_hex_mark_blocks superblock 27 ic_hex_mark_blocks superblock 26 ic_hex_mark_blocks superblock 31 ic_hex_mark_blocks superblock 32
61 ic_hex_mark_blocks numbers 33 53 edge_neighbors ic_hex_mark_blocks numbers 53 59 edge_neighbors ic_hex_mark_blocks numbers 59 37 edge_neighbors ic_hex_ogrid 1 m PART_1_1_1 INLET TWALL OUTLET BWALL NWALL PTWALL PBWALL V_ ,→ FLUID GEOM -version 50 ic_hex_mark_blocks unmark ic_hex_mark_blocks unmark ic_hex_move_node 68 VERT18 ic_hex_move_node 67 pnt.06 ic_hex_move_node 66 pnt.05 ic_hex_move_node 54 VERT17 ic_hex_move_node 69 pnt.03 ic_hex_move_node 73 pnt.04 ic_hex_move_node 72 VERT19 ic_hex_get_node_location { 66 } _tempx _tempy _tempz ic_hex_set_node_location x {$_tempx} -csys global node_numbers {{ 65 }} ic_hex_get_node_location { 54 } _tempx _tempy _tempz ic_hex_set_node_location x {$_tempx} -csys global node_numbers {{ 53 }} ic_hex_get_node_location { 72 } _tempx _tempy _tempz ic_hex_set_node_location x {$_tempx} -csys global node_numbers {{ 71 }} ic_hex_get_node_location { 66 } _tempx _tempy _tempz ic_hex_set_node_location y {$_tempy} -csys global node_numbers {{ 48 } { 47 ,→ }} ic_hex_find_comp_curve EDGEE5805 ic_hex_set_edge_projection 66 54 0 1 EDGEE5805 ic_hex_project_to_surface 66 54 ic_hex_set_edge_projection 66 67 0 1 EDGEE5805
62 ic_hex_project_to_surface 66 67 ic_hex_set_edge_projection 67 68 0 1 EDGEE5805 ic_hex_project_to_surface 67 68 ic_hex_find_comp_curve EDGEE6005 ic_hex_set_edge_projection 68 69 0 1 EDGEE6005 ic_hex_project_to_surface 68 69 ic_hex_set_edge_projection 69 73 0 1 EDGEE6005 ic_hex_project_to_surface 69 73 ic_hex_set_edge_projection 72 73 0 1 EDGEE6005 ic_hex_project_to_surface 72 73 ic_hex_find_comp_curve EDGEE5905 ic_hex_set_edge_projection 60 72 0 1 EDGEE5905 ic_hex_project_to_surface 60 72 ic_hex_set_edge_projection 54 60 0 1 EDGEE5905 ic_hex_project_to_surface 54 60 ic_hex_mark_blocks unmark ic_hex_mark_blocks superblock 22 ic_hex_mark_blocks superblock 27 ic_hex_mark_blocks superblock 32 ic_hex_change_element_id VORFN ic_delete_empty_parts ic_set_global geo_cad 0.008 toler ic_geo_new_family GEOM ic_boco_set_part_color GEOM ic_point {} PART_1_1_1 pnt.07 VERT18+vector(0,0.04,0) ic_point {} NWALL pnt.08 pnt.06+vector(-0.04,0.04,0) ic_point {} NWALL pnt.09 pnt.05+vector(-0.04,0,0)
63 ic_point {} PTWALL pnt.10 pnt.03+vector(0,0.04,0) ic_point {} PTWALL pnt.11 pnt.04+vector(0.04,0.04,0) ic_point {} PART_1_1_1 pnt.12 VERT19+vector(0.04,0,0) ic_hex_mark_blocks unmark ic_hex_mark_blocks unmark ic_hex_split_grid 42 67 pnt.08 m PART_1_1_1 INLET TWALL OUTLET BWALL NWALL ,→ PTWALL PBWALL V_FLUID GEOM VORFN ic_hex_move_node 81 pnt.07 ic_hex_move_node 80 pnt.08 ic_hex_move_node 78 pnt.09 ic_hex_move_node 82 pnt.10 ic_hex_move_node 83 pnt.11 ic_hex_move_node 79 pnt.12 ic_hex_get_node_location { 79 } _tempx _tempy _tempz ic_hex_set_node_location x {$_tempx} -csys global node_numbers {{ 77 }} ic_hex_get_node_location { 78 } _tempx _tempy _tempz ic_hex_set_node_location x {$_tempx} -csys global node_numbers {{ 76 }} ic_hex_link_shape 78 80 66 67 1.0 ic_hex_link_shape 80 81 67 68 1.0 ic_hex_link_shape 81 82 68 69 1.0 ic_hex_link_shape 82 83 69 73 1.0 ic_hex_link_shape 79 83 72 73 1.0 ic_hex_renew ic_hex_set_mesh_params PART_1_1_1 INLET TWALL OUTLET BWALL NWALL PTWALL ,→ PBWALL V_FLUID GEOM -version 110 ic_hex_renew
64 ic_hex_set_mesh_params PART_1_1_1 INLET TWALL OUTLET BWALL NWALL PTWALL ,→ PBWALL V_FLUID GEOM -version 110 ic_set_meshing_params curve {EDGEE5405 EDGEE5505 EDGEE5605 EDGEE5705 EDGEE ,→ 5805 EDGEE5905 EDGEE6005} emax 0.009 emin 0 ehgt 0 edev 0 hrat 0 ewid ,→ 0 nlay 0 prism_height_limit 0 law -1 ic_hex_set_mesh 11 47 n 75 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 default ,→ copy_to_parallel unlocked ic_hex_set_mesh 11 47 n 75 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 default ,→ copy_to_parallel unlocked ic_hex_set_mesh 47 41 n 17 h1rel 0.535376959846 h2rel 0.0 r1 2 r2 2 lmax 1e ,→ +10 default copy_to_parallel unlocked ic_hex_set_mesh 47 41 n 175 h1rel 0.535376959846 h2rel 0.0 r1 2 r2 2 lmax 1 ,→ e+10 default copy_to_parallel unlocked ic_hex_set_mesh 47 41 n 175 h1rel 0.535376959846 h2rel 0.0 r1 2 r2 2 lmax 1 ,→ e+10 default copy_to_parallel unlocked ic_hex_set_mesh 41 13 n 175 h1rel 0.415177894737 h2rel 0.0 r1 2 r2 2 lmax 1 ,→ e+10 default copy_to_parallel unlocked ic_hex_set_mesh 41 13 n 175 h1rel 0.415177894737 h2rel 0.0 r1 2 r2 2 lmax 1 ,→ e+10 default copy_to_parallel unlocked ic_hex_set_mesh 13 34 n 250 h1rel 0.0 h2rel 0.237434166667 r1 2 r2 2 lmax 1 ,→ e+10 default copy_to_parallel unlocked ic_hex_set_mesh 13 34 n 250 h1rel 0.0 h2rel 0.237434166667 r1 2 r2 2 lmax 1 ,→ e+10 default copy_to_parallel unlocked ic_hex_set_mesh 34 56 n 150 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 ,→ default copy_to_parallel unlocked ic_hex_set_mesh 34 56 n 150 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 ,→ default copy_to_parallel unlocked
65 ic_hex_set_mesh 56 62 n 2 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 default ,→ copy_to_parallel unlocked ic_hex_set_mesh 56 62 n 250 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 ,→ default copy_to_parallel unlocked ic_hex_set_mesh 56 62 n 250 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 ,→ default copy_to_parallel unlocked ic_hex_set_mesh 62 38 n 1725 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 ,→ default copy_to_parallel unlocked ic_hex_set_mesh 62 38 n 1755 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 ,→ default copy_to_parallel unlocked ic_hex_set_mesh 62 38 n 1755 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 ,→ default copy_to_parallel unlocked ic_hex_set_mesh 62 38 n 175 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 ,→ default copy_to_parallel unlocked ic_hex_set_mesh 38 21 n 450 h1rel 0.0165079047619 h2rel 1.0 r1 2 r2 2 lmax ,→ 1e+10 default copy_to_parallel unlocked ic_hex_set_mesh 38 21 n 450 h1rel 0.0165079047619 h2rel 1.0 r1 2 r2 2 lmax ,→ 1e+10 default copy_to_parallel unlocked ic_hex_set_mesh 33 76 n 150 h1rel 0.0 h2rel 0.292948009052 r1 2 r2 2 lmax 1 ,→ e+10 default copy_to_parallel unlocked ic_hex_set_mesh 33 76 n 150 h1rel 0.0 h2rel 0.292948375237 r1 2 r2 2 lmax 1 ,→ e+10 default copy_to_parallel unlocked ic_hex_set_mesh 76 65 n 95 h1rel 0.0 h2rel 0.0 r1 2 r2 2 lmax 1e+10 default ,→ copy_to_parallel unlocked ic_hex_set_mesh 76 65 n 95 h1rel 0.0 h2rel 2.5e-005 r1 2 r2 2 lmax 1e+10 ,→ geo2 copy_to_parallel unlocked
66 ic_hex_create_mesh PART_1_1_1 INLET TWALL OUTLET BWALL NWALL PTWALL PBWALL ,→ V_FLUID proj 2 dim_to_mesh 3 nproc 8 ic_hex_write_file {C:/Users/jainanup/Final Mesh/0.65 million_optimization_ ,→ geometry_files/dp0/ICM/ICEMCFD/hex.uns} PART_1_1_1 INLET TWALL OUTLET ,→ BWALL NWALL PTWALL PBWALL V_FLUID GEOM proj 2 dim_to_mesh 3 no_boco ic_uns_load C:/Users/jainanup/FINALM~1/0C3AE~1.65M/dp0/ICM/ICEMCFD/hex.uns ,→ 3 0 {} 1 ic_uns_update_family_type visible {INLET NWALL GEOM V_FLUID BWALL PBWALL ,→ TWALL PTWALL ORFN OUTLET PART_1_1_1} {!NODE !LINE_2 QUAD_4} update 0 ic_boco_solver ic_boco_clear_icons ic_undo_group_begin ic_boco_solver {ANSYS Fluent} ic_solver_mesh_info {ANSYS Fluent} ic_undo_group_end ic_boco_solver ic_boco_solver {ANSYS Fluent} ic_solution_set_solver {ANSYS Fluent} 1 ic_boco_save {C:/Users/jainanup/Final Mesh/0.65 million_optimization_ ,→ geometry_files/dp0/ICM/ICEMCFD/ICM.fluentAnsys.fbc} ic_boco_save_atr {C:/Users/jainanup/Final Mesh/0.65 million_optimization_ ,→ geometry_files/dp0/ICM/ICEMCFD/ICM.fluentAnsys.atr} ic_set_global vid_options 1.0 wb_import_transfer_file_scale ic_delete_empty_parts ic_delete_empty_parts ic_save_tetin ICM.tin 0 0 {} {} 0 0 1 ic_uns_check_duplicate_numbers
67 ic_save_unstruct ICM.uns 1 {} {} {} ic_uns_set_modified 1 ic_hex_save_blocking ICM.blk ic_boco_solver ic_boco_solver {ANSYS Fluent} ic_solution_set_solver {ANSYS Fluent} 1 ic_boco_save ICM.fbc ic_boco_save_atr ICM.atr ic_save_project_file ICM.prj ic_exec {C:/Program Files/ANSYS Inc/v191/icemcfd/win64_amd/icemcfd/output- ,→ interfaces/fluent6} -dom {C:/Users/jainanup/Final Mesh/0.65 million_ ,→ optimization_geometry_files/dp0/ICM/ICEMCFD/ICM.uns} -b ICM.fbc -dim2 ,→ d -bin {C:/Users/jainanup/Final Mesh/0.65 million_optimization_ ,→ geometry_files/dp0/ICM/ICEMCFD/ICM} ic_uns_num_couplings ic_undo_group_begin ic_uns_create_diagnostic_edgelist 1 ic_uns_diagnostic subset all diag_type uncovered fix_fam FIX_UNCOVERED diag ,→ _verb {Uncovered faces} fams {} busy_off 1 quiet 1 ic_uns_create_diagnostic_edgelist 0 ic_undo_group_end ic_uns_min_metric Quality {} {}
68 REFERENCES Anderson, John. Computational Fluid Dynamics : The Basics With Applications. New York: McGraw-Hill, 1995, 1st ed. Print. Beachkofski, Brian and Grandhi, Ramana. “Improved Distributed Hypercube Sampling.” 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2002. Chen, Xuyong, Zhao, Lifeng, Ma, Jiaqing, and Liu, Yuansen. “Aerodynamic simulation of evacuated tube maglev trains with different streamlined designs.” Journal of Modern Transportation 20 (2012).2: 115–120. Dhert, Tristan, Ashuri, Turaj, and Martins, Joaquim R. R. A. “Aerodynamic shape optimization of wind turbine blades using a Reynolds-averaged Navier-Stokes model and an adjoint method.” Wind Energy 20 (2017).5: 909–926. Draper, N. and Smith, H. Applied Regression Analysis. New York, NY: Wiley, 1998, 3rd ed. F. R. Menter, M. Kuntz and Langtry, R. “A Comprehensive Study on Surface Roughness in Machining of AISI D2 Hardened Steel.” Advanced Materials Research 576 (2012): 60–63. Forrester, Alexander I.J. and Keane, Andy J. “Recent advances in surrogate-based optimization.” Progress in Aerospace Sciences 45 (2009).1-3: 50–79. Friendship, Systems. “CAESES.” 2019. URL https://www.caeses.com Galindo, J., Hoyas, S., Fajardo, P., and Navarro, R. “Set-Up Analysis and Optimization of CFD Simulations for Radial Turbines.” Engineering Applications of Computational Fluid Mechanics 7 (2013).4: 441–460. Garg, Nitin, Kenway, Gaetan K.W., Martins, Joaquim R.R.A., and Young, Yin Lu. “High-fidelity multipoint hydrostructural optimization of a 3-D hydrofoil.” Journal of Fluids and Structures 71 (2017): 15–39. Giunta, Anthony, Wojtkiewicz, Steven, and Eldred, Michael. “Overview of Modern Design of Experiments Methods for Computational Simulations (Invited).” 41st Aerospace Sciences Meeting and Exhibit. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2003. Gray, Justin S., Mader, Charles A., Kenway, Gaetan K. W., and Martins, Joaquim R. R. A. “Modeling Boundary Layer Ingestion Using a Coupled Aeropropulsive Analysis.” Journal of Aircraft 55 (2018).3: 1191–1199. Hyun, S, Kim, C, Son, J.H, Shin, S.H, and Kim, Y.S. “An efficient shape optimization method based on FEM and B-spline curves and shaping a torque converter clutch disk.” Finite Elements in Analysis and Design 40 (2004).13-14: 1803–1815.
69 Inc., Fluent. “6.3.5. Mass Flow Inlet Boundary Conditions.” 2006. URL https://www.sharcnet.ca/Software/Ansys/16.2.3/en-us/help/flu{_}ug/ flu{_}ug{_}sec{_}bc{_}mfinlet.html Incorporation, Fluent. “23.3.5 Energy Equation.” 2006. URL https://www.sharcnet.ca/Software/Fluent6/html/ug/node885.html Journel, A G and Huijbregts, C J. Mining Geostatistics. United Kingdom, 1976. Web. Kim, Tae-Kyung, Kim, Kyu-Hong, and Kwon, Hyeok-Bin. “Aerodynamic characteristics of a tube train.” Journal of Wind Engineering and Industrial Aerodynamics 99 (2011).12: 1187–1196. Kleijnen, Jack P.C. “Kriging metamodeling in simulation: A review.” European Journal of Operational Research 192 (2009).3: 707–716. Knudsen, Martin and Partington, J. R. “The Kinetic.Theoryof Gases. Some Modern Aspects.” The Journal of Physical Chemistry 39 (1934).2: 307–307. Launder, B. “Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc.” International Communications in Heat and Mass Transfer 1 (2003).2: 131–137. Menter, F. “Zonal Two Equation k-w Turbulence Models For Aerodynamic Flows.” 23rd Fluid Dynamics, Plasmadynamics, and Lasers Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1993. Menter, F. R. “Two-equation eddy-viscosity turbulence models for engineering applications.” AIAA Journal 32 (1994).8: 1598–1605. NASA. Langley Research Center Turbulence Modeling Resource. NASA, 2019. URL https://turbmodels.larc.nasa.gov O’Hagan, A. “Curve Fitting and Optimal Design for Prediction.” Journal of the Royal Statistical Society: Series B (Methodological) 40 (1978).1: 1–24. Opgenoord, Max M. J. and Caplan, Philip C. “Aerodynamic Design of the Hyperloop Concept.” AIAA Journal 56 (2018).11: 4261–4270. Panton, Ronald. Incompressible Flow. Hoboken, New Jersey: John Wiley and Sons, Inc, 2013, 4th ed. Print. Putra Adnan, F and Hartono, Firman. “Design of Single Stage Axial Turbine with Constant Nozzle Angle Blading for Small Turbojet.” Journal of Physics: Conference Series 1005 (2018): 012026.
70 Queipo, Nestor V., Haftka, Raphael T., Shyy, Wei, Goel, Tushar, Vaidyanathan, Rajkumar, and Kevin Tucker, P. “Surrogate-based analysis and optimization.” Progress in Aerospace Sciences 41 (2005).1: 1–28. Rasmussen, Carl Edward. Gaussian Processes in Machine Learning. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004, 63–71. Salter, R. M. “The Very High Speed Transit System.” Tech. rep., RAND Corporation, Santa Monica, CA, 1972. URL https://www.rand.org/pubs/papers/P4874.html Simpson, T.W., Poplinski, J.D., Koch, P. N., and Allen, J.K. “Metamodels for Computer-based Engineering Design: Survey and recommendations.” Engineering with Computers 17 (2001).2: 129–150. SpaceX. “Hyperloop Alpha.” 2013. URL https://www.spacex.com/sites/spacex/files/hyperloop{_}alpha.pdf Wang, Haipeng, Zhang, Bo, and Qiu, Qinggang. “Numerical study of the effects of wind shear coefficients on the flow characteristics of the near wake of a wind turbine blade.” Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 230 (2016).1: 86–98. Wilcox, D. C. “Formulation of the k-w Turbulence Model Revisited.” AIAA Journal 46 (2008).11: 2823–2838. Wilcox, David. Turbulence Modeling for CFD. La Cãnada, Calif: DCW Industries, 1998. Print. Wild, Stefan M., Regis, Rommel G., and Shoemaker, Christine A. “ORBIT: Optimization by Radial Basis Function Interpolation in Trust-Regions.” SIAM Journal on Scientific Computing 30 (2008).6: 3197–3219. Yang, Yi, Wang, Haiyang, Benedict, Moble, and Coleman, David. “Aerodynamic Simulation of High-Speed Capsule in the Hyperloop System.” 35th AIAA Applied Aerodynamics Conference. Reston, Virginia: American Institute of Aeronautics and Astronautics, 2017. Yu, Yin, Lyu, Zhoujie, Xu, Zelu, and Martins, Joaquim R.R.A. “On the influence of optimization algorithm and initial design on wing aerodynamic shape optimization.” Aerospace Science and Technology 75 (2018): 183–199. Zhang, Yaoping. “Numerical simulation and analysis of aerodynamic drag on a subsonic train in evacuated tube transportation.” Journal of Modern Transportation 20 (2012).1: 44–48.
71 BIOGRAPHICAL SKETCH Mr. Anup Jain was born in Dhule, India in 1994. After completing his high-school studies in Dhule, he pursued his undergraduate studies at Savitribai Phule Pune University, Pune, India, where he received his Bachelor of Engineering in the field of Mechanical Engineering. He worked for Atlas Copco as an Design Engineer Intern. In August 2017, he came to the University of Florida to pursue his Master of Science degree in mechanical engineering. During his masters he also worked as a Software Testing Intern (Spring 2019) at ANSYS INC.,MA in the Design Buisness Unit and as CFD Testing Intern (Fall 2019) at ANSYS INC.,NH in the Fluid Buisness Unit.
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