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A 3-D Viscous Vorticity Equation (VISVE) Method Applied to Flow Past a Hydrofoil of Elliptical Planform and a Propeller

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A 3-D Viscous Vorticity Equation (VISVE) Method Applied to Flow Past a Hydrofoil of Elliptical Planform and a Propeller Sixth International Symposium on Marine Propulsors smp’19, Rome, Italy, May 2019 A 3-D VIScous Vorticity Equation (VISVE) Method Applied to Flow Past a Hydrofoil of Elliptical Planform and a Propeller Chunlin Wu, Spyros A. Kinnas Ocean Engineering Group, Department of Civil, Architectural and Environmental Engineering The University of Texas at Austin, Austin, Texas 78712, USA ABSTRACT difference method. Both the differential approach A VIScous Vorticity Equation (VISVE) method that solves (Poisson’ equation) and the integral approach (Biot-savart the vorticity equation is proposed in this paper to simulate law) to recovering the solenoidal velocity field were the 3-D viscous flow past a hydrofoil or a propeller. The studied. Hansen et al. (2003) solved the 3-D vorticity spatial concentration of vorticity leads to a local solver equation in the cylindrical coordinates. The vorticity with a significantly small computational domain and a equation was also discretized by using the finite difference small number of cells, which results in much less method. A conjugate gradient method was adopted to solve computation time. The VISVE method is applied to a 3-D the Poisson’s equation. The method was applied to the hydrofoil of elliptical planform. Calculated results translating flow and rotating flow in a pipe. Meitz and provided good agreement with results from Naiver-Stokes Fasel (2000) expanded the vorticity method into the (N-S) methods, which validates the method. Also, some Fourier space by the Fourier series to simulate the preliminary results of the flow past the 5-bladed NSRCD transitional and turbulent channel flow. Lo et al. (2005) 4381 propeller are shown. applied the method to the 3-D lid-driven cavity flow at Keywords Reynolds numbers ranging from 100 to 2000. Vorticity equation, 3-D hydrofoil, Propeller, Viscous flow. For the existing applications of mesh-based 3-D vorticity 1 INTRODUCTION methods, the computational domains are restricted to be The 3-D vortex methods were successful in predicting the quite simple since the boundary conditions contain mixed development of vortex in the absence of wall boundaries at derivatives which are difficult to be implemented in high Reynolds numbers when the viscous effect is small. general domains. In the proposed VISVE method, the 3-D Generally, there are two types of mesh-free 3-D vortex vorticity equation is discretized in space by using the Finite methods, which are the vortex particle methods (Rehbach, Volume Method (FVM). A Poisson’s equation is solved 1978; Saffman, 1981) and the vortex filament methods based on the finite volume discretization to recover the (Chorin, 1981; Leonard, 1985). When meshes are partially divergence-free velocity field. Moreover, a vorticity used for solving the velocity, another type of 3-D vorticity creation scheme is utilized to determine the amount of method evolved from the above two methods. This method vorticity created by enforcing the no-slip and non- is known as the Vortex In Cell (VIC) method (Tryggvason penetrating boundary conditions on the wall. The proposed et al., 1991; Almgren et al., 1994). For the above 3 vorticity creation scheme can treat the general domain methods, most of the existing literature neglect the effect boundaries, which significantly extends the scope of of diffusion by assuming a high Reynolds number. Also, applications of the current method. extensive studies have been conducted to include viscous The VIScous Vorticity Equation (VISVE) method was first corrections into those methods (Degond & Mas-Gallic, proposed by Tian & Kinnas (2015) as an alternative 1989; Leonard, 1980). Actually, the viscous diffusion numerical tool to predict the performance of propeller at affects the evolution of vortex significantly, especially at on- and off-design conditions. However, the method needs moderate and low Reynolds numbers. In addition, the systematic validations and further improvements. Li & viscosity is responsible for the production of vorticity on Kinnas (2017) applied the scheme to the unidirectional and the wall. Therefore, the viscosity of fluids is essential in alternating flows past a 2-D cylinder. Xing & Kinnas describing the interactions between fluid and solid near the (2018) investigated the cavitating flow around a 2-D wall surface. hydrofoil by including the mixture cavitation model into When the full mesh is used, the method is usually referred VISVE. Wu & Kinnas (2018) extended the VISVE method to as the viscous vorticity method. Wu et al. (1995) to 3-D hydrofoils considering the advection flow, in which proposed a 3-D vorticity method in Cartesian coordinates. only the advection and stretching terms were included in The vorticity equation was discretized using the finite the vorticity equation. One year later, Wu & Kinnas (2019) incorporated the diffusion term into the vorticity equation in space and is discretized using the Euler backward with skewness corrections and applied the improved scheme in time. The discretized vorticity equation is shown scheme to fully viscous flow past a hydrofoil in non- in Equation (4). orthogonal grids. Reliable results were obtained. The 휔푛+1 − 휔푛 VISVE is also hybridly paralleled using both Open-MP 퐴 푖 푖 + ∑(흎푛+1 × 풗푛) ⋅ 훥풍 푖 ∆푡 and MPI to improve efficiency (Wu & Kinnas, 2018), 휕퐴 (4) where VISVE was shown to require significantly less = −푣 ∑(훻 × 흎푛+1) ⋅ 훥풍 computing time than N-S, due primarily to the significantly 휕퐴 smaller grid in VISVE. The convective & stretching term and the diffusive term The VISVE method has several advantages in applying to are further discretized, and the scheme is explained simulate the flow past a propeller: thoroughly by Tian (2014). A hybrid interpolation scheme (a) It is a local solver with a significantly small that couples the first order upwind method, the second computational domain and a small number of cells, which order upwind method and the Quadratic Upstream results in much less computation time. Interpolation for Convective Kinematics (QUICK) method is adopted in this work (Woodfield, 2004). (b) The effect of other blades is considered in an iterative manner, which is similar to the method used in PROPCAV 2.3 Velocity Calculation and MPUF-3A. Only the flow on the key blade needs to be The velocity field 풗 is directly related to the potential modeled. For axisymmetric case: duplicate the variables of vector 흍 the key blade. For non-axisymmetric case: use the ∇ × 흍 = 풗 (5) variables of the key blade at different blade angles. Considering the relationship between velocity and vorticity (c) The method allows for easily built grids. Since only the (흎 = ∇ × 풗), we further have vicinity of the propeller needs to be modeled without using ∇2흍 = −흎 (6) the periodic boundary conditions, the grid generation can be significantly simplified. Actually, this process can be Equation (6) is a Poisson’s equation, which can be solved made automated. through a direct volume integral of the Green’s function. 2 METHODOLOGY 흎(풙) 흍(풙푐) = ∫ 푑푉 + 흍푖푛 (7) 2.1 Velocity Decomposition 푽 4휋|풙 − 풙푐| The total velocity field 풗 is composed of two parts, which where 풙푐 is the point where 흍 is calculated, 흎(풙) is the are the background flow velocity 푼 and the vorticity 푖푛 vorticity distribution in the computational domain. 흍푖푛 is induced velocity 풖 the stream-function vector of the background inflow. The 풗 = 푼푖푛 + 풖 (1) direct integral of Equation (7) is very expensive to conduct in 3-D space. For all the points where velocity is evaluated, For a given vorticity distribution, the corresponding the inductions from all the cells with significant vorticity velocity field can be uniquely determined according to the should be considered, which is about 푁2 complexity where Helmholtz velocity decomposition. Therefore, the 푁 is the number of cells. decomposition in Equation (1) is sole for a known flow field. In order to improve the computational efficiency, an FVM scheme is adopted to solve the Poisson’s equation. With 2.2 Vorticity Transport Equation proper implementation, the computational complexity can In the incompressible viscous flow, the Navier-Stokes (N- be reduced to be almost proportional to 푁. S) equation in the Eulerian frame reads 휕풗 ∇푝 Consider a known velocity field 풘 which conserves the + (풗 ∙ ∇)풗 = − + 휈∇2풗 (2) vorticity in the flow field. 휕푡 휌 흎 = ∇ × 풘 (8) where 풗 represents the velocity vector, 푝 is the pressure, 푡 is the time, 휌 is the density of the fluid and 휈 is the However, this vorticity preserving velocity 풘 does not kinematic viscosity of the fluid. The vorticity transport necessarily satisfy the continuity equation. Therefore, to equation could be obtained by taking curl on both sides of enforce the divergence-free condition of the velocity field, Equation (2), and the curl form of vorticity equation can be a potential correction 휂 is introduced to correct the velocity derived along the way as field 풘. 휕흎 풖 = 풘 + ∇휂 (9) + ∇ × (흎 × 풒) = −휈∇ × (∇ × 흎) (3) 휕푡 By taking the divergence on both sides of Equation (9), the where 흎 = ∇ × 풗 is the vorticity. The biggest advantage of Poisson’ equation for 휂 can be obtained the curl formulation of the vorticity equation is that the ∇2휂 = −∇ ∙ 풘 (10) divergence-free condition of vorticity is automatically satisfied in time. Equation (3) is discretized using the Finite Therefore, the volume integral Equation (7) is only used to Volume Method (FVM) by applying the Stokes’ theorem calculate the Neumann boundary conditions for the above Poisson’s equation. The calculation of vorticity preserving velocity field 풘 is The pressure at a reference point on the wall can be not unique. Recall the discretized vorticity equation (4), it determined by Equation (14) and (15), and the pressure at is not hard to find that other locations on the wall can be calculated based on 풘푛+1 = −훥푡[흎푛+1 × 풗푛 + 휈훻 × 흎푛+1] + 풗푛 (11) Equation (16).
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