1 Evaluation of an Alternate Incompressible Energy Enstrophy Turbulence Model
Douglas F. Hunsaker Utah State University
A concise overview of turbulence modeling challenges is laminar flow fields and are only available for the most presented along with developmental concerns of rudimentary geometries. Therefore, turbulent flow modeling traditional turbulence models. An alternate energy is left almost entirely to the mercy of CFD. enstrophy turbulence model developed by Dr. Warren The governing principles of mass and momentum Phillips is given followed by plans for closing the new conservation apply to turbulent flow and can be employed model and evaluating the subsequent closure directly through CFD methods. However, the grid coefficients. This paper is effectively an overview of a refinement required to capture the small scales of turbulence PhD dissertation proposal and includes only a brief using CFD techniques is so extreme that solutions based on description of the project outline. this technique can be fairly computationally expensive. Such CFD techniques are called Direct Numerical I. INTRODUCTION Simulation (DNS) methods and are currently employed mainly for relatively small domains in which the small Fluid mechanics has been a topic of study for hundreds scales of turbulence are large in comparison to the grid of years and has fascinated many of the greatest minds of element size of the domain of interest. For a concise history. Many applications today are dependent on correctly discussion of DNS, see Bernard and Wallace (2002a) or understanding and predicting the motion of fluids and much Wilcox (2006a). For a more extensive review of the subject research has been conducted to that end. The laws that including references to recent work, see Moin and Mahesh govern fluid motion have been understood since the mid (1998). 1800s (Navier 1823; Stokes 1845) and the vector In order to model the turbulent characteristics of larger mathematics needed to fully analyze three-dimensional fluid geometries using CFD, supplementary relationships are mechanics was sufficiently understood only a few decades commonly employed to the governing equations of fluid later (Hamilton 1844, 1853, 1866; Boyer 1989). Since that motion. These additional relationships should be based on time, these laws of motion and mathematical properties have the physics of turbulent motion to provide accurate been studied by countless researchers, and much progress in simulations. The physical characteristics of turbulence have the physical understanding of both laminar and turbulent been studied in detail during the past century. Much of our flows has been achieved. However, the complexity current understanding of turbulence has been constructed presented in completely understanding and correctly through the outcomes of countless experiments and has predicting fluid mechanics leaves room for much resulted in much progress. Although these findings have improvement in the field. greatly expanded our understanding of turbulent flow Many analytical solutions exist for problems with characteristics, they are somewhat preliminary in nature and simple geometries because the governing equations can be have not proven to be fully representative of turbulent simplified and analytically applied. However, for more behavior. In other words, there is still much to be learned complex geometries and flow fields, the boundary about turbulence. To this extent, notable authors have conditions and vector equations are quite complex and commented. computational means must be employed. Computational Fluid Dynamics (CFD) is a method of fluid flow calculation “I am an old man now, and when I die and go to that uses a gridded domain to predict the flow field in a Heaven there are two matters on which I hope for given geometry. This modeling method has grown in enlightenment. One is quantum electrodynamics popularity as computational power has increased, making and the other is the turbulent motion of fluids. And solutions to complex flow problems more readily about the former I am rather optimistic.” -Sir achievable. However, even with modern computational Horace Lamb (Goldstein 1969) power, accurate solutions to flow problems can take days or “Turbulence was probably invented by the Devil weeks to converge. on the seventh day of Creation when the Good The difficulty of modeling fluid mechanics is greatly Lord wasn't looking.” -P. Bradshaw (1994) increased when turbulent flow is considered. Indeed, the most intriguing and complex flow solutions are those for “… less is known about the fine scale turbulence … turbulent flow fields. Analytical solutions for turbulent flow than about the structure of atomic nuclei. Lack of fields are much more difficult to develop than those for basic knowledge about turbulence is holding back 2 progress in fields as diverse as cosmology, average flow field from turbulent fluctuations in unsteady meteorology, aeronautics and biomechanics.” -U. common turbulent flow fields. For an overview on the Frisch and S. Orszag (1990) subject see Bernard and Wallace (2002b). For a detailed discussion on averaging see Phillips (2008b) or Wilcox “Turbulence is the last great unsolved problem of (2006b). classical physics. Remarks of this sort have been The complex nature of turbulent flow is not yet fully variously attributed to Sommerfeld, Einstein, and understood. However, research conducted during the past Feynman, although no one seems to know precise century has allowed for certain properties of mean turbulent references, and searches of some likely sources flow to be quantified. Therefore, traditional governing have been unproductive. Of course, the allegation equations of mean flow have been developed over time. The is a matter of fact, not much in need of support by a challenge of forming equations truly representative of mean quotation from a distinguished author. However, it turbulent flow has inspired much research that has resulted would be interesting to know when the matter was in varying degrees of success. The most widely-used first recognized.” -P.J. Holmes, G. Berkooz, and resulting models for internal flows include the k ε model J.L. Lumley (1996) based on the development of Jones and Launder (1972), and
variations of the k ω model originally developed by Although the physical phenomenon of turbulence is not Kolmogorov (1942). Commonly used aerodynamic flow fully understood, the basic features of turbulent motion are models include a model developed by Spalart and Allmaras known. First, turbulence is caused by inertial forces (1992) and a model developed by Baldwin and Barth overpowering viscous forces within a fluid. At low (1990). Reynolds numbers, viscous forces dominate, producing Retracing the derivations of traditional turbulence laminar flow. However, as the Reynolds number increases, modeling equations reveals seemingly minor yet possibly the inertial forces of the fluid increase and eventually significant assumptions which have perhaps hindered the overcome the viscous forces. At this point, velocity and validity of the models in the past. Recent re-examination of pressure fluctuations develop and the flow becomes these equations by Phillips (2008c) has led to alternative irregular. This leads to the second most fundamentally developments of the well-known and explored k ε, k ω, and understood characteristic of turbulence. Turbulent flow is k ζ turbulence models. It is possible that the adjusted comprised of fluctuations in pressure, velocity, and equations will produce more accurate results for turbulent temperature making it impossible to reproduce the exact flow calculations. fluctuations from consecutive experiments although the This paper includes the development of an alternate average flow field is recreated. turbulence modeling approach suggested by Phillips A definition for turbulence commonly accepted today (2008d), and an overview of the proposed research to be was given by Hinze (1975): conducted by the author. “Turbulent fluid motion is an irregular condition of flow in which the various quantities show a II. PHILLIPS TURBULENCE CLOSURE random variation with time and space coordinates, A. Driving Concerns so that statistically distinct average values can be Wilcox discusses three major concerns with the manner discerned.” in which the traditional turbulence models previously
discussed are derived and closed. A definition that is perhaps more mathematically precise is 1. The dissipation per unit mass used in the traditional that given by Phillips (2008a): turbulence models is not the true dissipation of “Turbulent fluctuations are irregular variations in turbulent kinetic energy per unit mass (Wilcox 2006c). certain quantities of a flow field (such as pressure, 2. The length scale used to close the traditional turbulence temperature and velocity) that are not predictably models is the length associated with the smaller repeatable from one experiment to another.” turbulent eddies which have higher strain rates. However, the larger turbulent eddies are the energy- The main difference between the two definitions is that bearing eddies and are primarily responsible for the Hinze’s definition does not specify the averaging method transport of turbulent-kinetic energy in a fluid flow required to deduce the statistical information while Phillips’ field (Wilcox 2006d) definition specifies ensemble averaging as the suitable 3. The traditional closing transport equations are averaging method. It can be shown and is well understood developed by simple analogy and dimensional analysis that ensemble averaging must be used instead of spatial and and are not developed rigorously from the Navier- temporal averaging in order to accurately distinguish the Stokes equations (Wilcox 2006d) 3 Phillips (2008e) echoes these concerns and adds two Using this notation, the velocity vector and pressure scalar additional concerns to the list. at a point in the flow can be written 4. The traditional dissipation per unit mass actually ~ includes a portion of the total molecular transport V = V + V (4) term. Therefore, the traditional turbulent molecular and transport term neglects a portion of the molecular transport in the traditional turbulence models. p = p + ~p (5) 5. Because a portion of the molecular transport is neglected in the traditional turbulent molecular It is important to note that the average of the fluctuating transport term, subsequent application of the component is zero by definition. Boussinesq hypothesis to the molecular turbulent ~ V ≡ ~p ≡ 0 (6) transport is in question. These concerns affect the derivation of the turbulent- Applying Eq. (4) and Eq. (5) to the general form of the energy-transport equation, the derivation of the turbulent- mechanical energy equation and taking the ensemble dissipation-transport equation, and the accepted length and average of the resulting formulation gives velocity scales. The following sections supply a concise ∂ overview of alternate forms of the kinetic-energy-transport ρ[ ( 1 V 2 + k) + V ⋅∇( 1 V 2 + k) equation and the dissipation-transport equation as well as an ∂t 2 2 alternate turbulent length and frequency scale suggested by ~ ~ ~ 1 ~ + V ⋅∇(V ⋅V) + V ⋅∇( V 2 ]) Phillips. 2 ~ ~ = ∇⋅{ [∇( 1 V 2 + k) + (V ⋅∇)V + (V ⋅∇)V]} (7) B. Turbulent Energy Transport 2 ~ ~ The turbulent-kinetic-energy transport equation can be − V ⋅[∇pˆ − goZ∇ρ]− V ⋅∇pˆ v v v v derived from the specific Reynolds stress tensor. In contrast, v v v ~ v ~ the turbulent-energy transport equation can also be − 2 [ (S V)⋅ (S V) + (S V)⋅ (S V)] developed from the mechanical energy equation which is where k is defined as the specific turbulent kinetic energy formed by taking the dot product of the velocity vector with v the Navier-Stokes equations. Defining the total hydrostatic ~ τ k = 1 V 2 = 1 ~ 2 ~ 2 ~ 2 = − 1 trace (8) pressure as 2 2 Vx + Vy + Vz 2 ρ pˆ = p +ρ g Z + 2 ∇ ⋅ V (1) o 3 and the two pressure terms are the mean and fluctuating hydrostatic pressure terms respectively the Navier-Stokes equations can be written in vector form as v pˆ ≡ p + ρg Z + 2 ∇ ⋅ V (9) ∂V o 3 ρ + (V ⋅∇)V = −∇pˆ + ∇ ⋅ 2[ S(V)]+ goZ∇ρ (2) ∂t ~ 2 ~ pˆ ≡ ~p + ∇ ⋅ V (10) 3 Taking the dot product of the velocity vector with the Navier-Stokes equations, rearranging, and using Using these definitions for pressure terms, the mathematical identities gives the mechanical energy Reynolds-averaged Navier-Stokes equations in vector form equation for a Newtonian fluid can be written as ∂ ∂V ~ ~ 1 2 1 2 ρ + (V ⋅∇)V + (V ⋅∇)V ρ ( V ) + V ⋅∇( V ) ∂t 2 2 ∂t (11) v = ∇⋅{ [∇( 1 V 2 ) + (V ⋅∇)V]} ˆ 2 (3) = −∇p + ∇ ⋅ 2[ (S V)]+ goZ∇ρ
− V ⋅[∇pˆ − goZ∇ρ] v v Taking the dot product of the mean velocity vector with this −2 S(V)⋅S(V) form of the Reynolds-averaged Navier-Stokes equations, rearranging, and using mathematical identities gives the The velocity and pressure at any given time in a turbulent mean mechanical energy equation for a Newtonian fluid flow can be expressed as the sum of the ensemble average of the property at that point and the fluctuating component. In the following notation, an over-bar presents the ensemble average and a tilde represents a fluctuating component. 4
∂ 1 2 1 2 ~ ~ ∂k ρ ( V ) + V ⋅∇( V ) + V ⋅(V ⋅∇)V ρ + (V ⋅∇)k ∂t 2 2 ∂t v v 1 2 2 = ∇⋅{ [∇( V ) + (V ⋅∇)V]} (12) = 2 t (S V) ⋅ (S V) − (ρk + t∇ ⋅ V)∇ ⋅ V (17) 2 v v 3 v v ~ − V ⋅[∇pˆ − g Z∇ρ]− 2 (S V)⋅ (S V) − ρε + ∇ ⋅ ((ν +ν t σ k ){ρ∇k o v + 2 ∇(ρk + ∇ ⋅ V) − 2∇ ⋅[ S(V)]}) Applying mathematical identities to this equation, 3 t t subtracting the result from Eq. (7), and applying more Note that the dissipation term used in this differential mathematical identities gives the alternate form of the equation represents the exact dissipation per unit mass. turbulent-kinetic-energy-transport equation However, this equation can be used interchangeably with ∂k other turbulence models that include a modeled dissipation ρ + (V ⋅∇)k ∂t term. A close look at Eq. (17) reveals that the molecular v v transport term is not simply a pure gradient diffusion v v ~ v ~ ~ = τ ⋅ J(V) − 2 [ (S V) ⋅ (S V) − 1 (∇ ⋅ V)2 ] process as is assumed with traditional developments. 3 (13) v Therefore, it is probable that this version of the turbulent- ~ ~ v + p(∇ ⋅ V) + ∇ ⋅ ( ∇k −ν∇ ⋅ τ) energy-transport equation will be more accurate than the ~ ~ ~ ~ ~ − ∇ ⋅[ 1 ρV 2V + ~pV + 2 (∇ ⋅ V)V] traditional equations. 2 3 Before leaving the topic of turbulent-energy transport, it A close look at this equation reveals that the terms on is important to note that the turbulent-energy-transport the right-hand side are the true volumetric production, equation can alternatively be written in the form viscous dissipation, pressure dilation, molecular transport, ∂k and the volumetric turbulent transport of turbulent kinetic ρ + (V ⋅∇)k ∂t energy. Note that the only approximation used to develop v v this equation was that for a Newtonian fluid. Also note that = 2 (S V) ⋅ (S V) − 2 (ρk + ∇ ⋅ V)∇ ⋅ V − ω~2 t 3 t the assumption of constant dynamic viscosity was not made 1 in the development of this governing equation is done in − 4 ∇ ⋅ ({ ∇(ρk + t∇ ⋅ V) (18) v 3 traditional developments. Applying the Boussinesq v − ∇ ⋅[ tS(V)]} ρ) + ∇ ⋅ ((ν +ν t σ k ){ρ∇k hypothesis to this equation gives the Boussinesq-based v turbulent-energy-transport equation. + 2 ∇(ρk + ∇ ⋅ V) − 2∇ ⋅[ S(V)]}) 3 t t ∂k ρ + (V ⋅∇)k where the change of variables ∂t v v v v v v ~ ~ ~ 1 ~ 2 = 2 (S V)⋅ (S V) − 2 (ρk + ∇ ⋅ V)∇ ⋅ V ε ≡ 2ν[ (S V)⋅ (S V) − (∇⋅V) ] t 3 t 3 v v ~2 1 ~ ~ 1 ~ ~ =νω + 4ν∇⋅({ ∇(ρk + ∇⋅ V) (19) − 2 [ (S V)⋅ (S V) − (∇ ⋅ V)2 ] + ~p(∇ ⋅ V) (14) 3 t 3 v − ∇ ⋅[ S(V)]} ρ) + ∇ ⋅((ν +ν t σ k ){ρ∇k t v + 2 ∇(ρk + ∇ ⋅ V) − 2∇ ⋅[ S(V)]}) has been applied in favor of using the root-mean-square 3 t t (RMS) fluctuating vorticity instead of the exact dissipation Defining the dissipation per unit mass as term. The dilatational terms have been shown (Huang et al. v v 1995) to be negligibly small for mean velocities under Mach v ~ v ~ ~ ε~ ≡ 2ν[ (S V) ⋅ (S V) − 1 (∇ ⋅ V)2 ] (15) three and have been neglected in Eq. (18). The square of the 3 RMS fluctuating vorticity is defined as and neglecting the pressure dilation term ~2 ~ ~ ~ ω ≡ (∇ × V) ⋅ (∇ × V) (20) ~p(∇ ⋅ V) ≅ 0 (16) The advantage of applying this change of variables is gives a version of the turbulent-energy-transport equation two-fold. First, vorticity is a well-understood property of a that can be used in a traditional k ε or k ω model. fluid flow field while the turbulent dissipation is not. Second, the divergence of the vorticity of any flow field is always zero. This mathematical property could be beneficial in the development of a vorticity-based-transport equation 5 which is not available in the development of a turbulent- dissipation-transport equation. III. DISSERTATION PROJECT DESCRIPTION
A. Overview C. Turbulent Length Scale Phillips’ development of the governing turbulent- As previously mentioned, the length scale commonly energy-transport equation and turbulent-characteristic length employed in the development of traditional turbulence scale provide a sound alternative to current turbulence models is that associated with the turbulent dissipation. models. In order to assess the validity of the Phillips However, the larger turbulent eddies are the energy-bearing development, the suggested model must be closed and eddies and are responsible for the majority of the transport evaluated. This includes considering various closure of turbulent energy in a flow. An alternate length scale methods, determining appropriate values for the closure suggested by Phillips (2008d) is formulated by examining constants, and comparing numerical results to both the fluctuating vorticity of a flow field. It is important to experimental and numerical results of well-known cases. note that the angular velocity of a fluid element is one-half The proposed research will concentrate on evaluating the local vorticity. The mean kinetic energy per unit mass is methods for closing the Phillips vorticity-based turbulence traditionally defined as one-half the mean square of the model. translational velocity. Phillips suggests that the mean kinetic energy per unit mass can also be defined as one-half the B. Closure Methods mean square of the angular velocity multiplied by the square Several options exist for closing the Phillips k-vorticity of some length scale. This length scale should be a length model. A short explanation of these methods is presented scale associated with the turbulent energy. Therefore, here. ~ ~ ~ ~ k = 1 V ⋅ V = 1 [ 1 (∇ × V) ⋅ (∇ × V ]) l2 (21) 1. RMS Turbulent Vorticity Closure 2 2 4 k Perhaps the simplest approach to closing the Phillips where lk is the length scale. From the definition of the vorticity model is to model the RMS turbulent vorticity in fluctuating vorticity given in Eq. (20), Phillips defines this terms of the mean fluid velocity, the specific turbulent length scale as kinetic energy, and the turbulent eddy viscosity. The fluctuating vorticity RMS is defined by Eq. (20). By 8k lk ≡ ~ (22) analogy with Eq. (24), a turbulent-vorticity transport ω equation can be obtained. This results in a model defined by k D. Suggested k Vorticity Model ν t = Cν ~ (25) Using Eq. (22) as the length scale in the definition of ω the turbulent viscosity, an alternate k ω model can be ∂k + (V ⋅∇)k formed. Combining this new definition of turbulent ∂t viscosity with Eq. (18) gives the two fundamental equations v v v = 2ν (S V)⋅ (S V) −ν (ω~2 + 4∇ ⋅{1 ∇k − ∇ ⋅[ν S(V)]}) (26) of Phillips’ k ω model. t 3 t v k + ∇ ⋅((ν +ν σ ){ 5 ∇k−2∇ ⋅[ν S(V)]}) ν = C (23) t k 3 t t ν ω~ ∂ω~ + V ⋅∇ω~ ∂k ∂t + (V ⋅∇)k (27) ∂t ~ v v v v v ω ~2 ~ v v v = 2C ~ ν (S V) ⋅ (S V) − C ~ ω + ∇ ⋅[(ν +ν σ ~ )∇ω] = 2ν (S V)⋅ (S V) −ν (ω~2 + 4∇ ⋅{1 ∇k − ∇ ⋅[ν S(V)]}) (24) ω1 t k ω 2 t ω t 3 t v 5 where Cν, σk, Cω~1 , Cω~2 , and σ ω~ are the closure constants + ∇ ⋅((ν +ν t σ k ){ ∇k − 2∇ ⋅[ν tS(V)]}) 3 for the model and need to be evaluated. This formulation can be directly recast in terms of where Cν and σk are closure constants. The most important contributions of the Phillips enstrophy. The enstrophy is defined as the solenoidal development are first, the inclusion of the last term in Eq. dissipation divided by the molecular viscosity. Additionally, (24) and second, the proposition of using the local energy- the enstrophy is equal to the square of the fluctuating weighted turbulent length scale found in Eq. (22) to define vorticity. the eddy viscosity. These contributions have not been εˆ ~ 2 included in other turbulence models and could potentially ζ ≡ ≡ ω (28) ν improve RANS turbulence modeling accuracy. 6 v v Using this change of variables, the formulation can be ∂ζ ζ v v + V ⋅∇ζ = 2C ν (S V)⋅ (S V) written ∂t ζ 1 t k (36) 1 2 ζ 2 ν t = Cν k ζ (29) − C ν + ∇ ⋅[(ν +ν σ )∇ζ ] ζ 2 k t ζ ∂k + (V ⋅∇)k ∂t where Cν, σk, Cζ1, Cζ2, and σζ are the closure constants for v v v the model and may need to be reevaluated because the = 2ν (S V)⋅ (S V) −ν (ζ + 4∇ ⋅{1 ∇k − ∇ ⋅[ν S(V)]}) (30) t 3 t solenoidal dissipation, εˆ, differs from the traditional v 5 definition of dissipation, ε . + ∇ ⋅((ν +ν t σ k ){ ∇k−2∇ ⋅[ν tS(V)]}) 3 3. DNS Solenoidal Dissipation Closure ∂ζ Another possibility to closing the Phillips vorticity- + V ⋅∇ζ ∂t based model is to use the results of a recently-developed v v DNS-based solenoidal-dissipation model by Kreuzinger, ζ 3 2 = 4C ~ ν (S V)⋅ (S V) − 2C ~ ζ (31) Friedrich, and Gatski (2006) that has provided good ω1 t k ω 2 agreement with DNS results. The solenoidal-dissipation- 1 + ∇ ⋅[(ν +ν σ ~ )∇ζ ] − (ν +ν σ ~ )∇ζ ⋅∇ζ ζ t ω 2 t ω transport equation for incompressible flow is given in this study as where the closure coefficients are the same as those needed ∂εˆ for Eqs. (25)– (27). + V ⋅ ∇εˆ ∂t 2. Solenoidal Dissipation Closure v v εˆ v v Another approach to closing the formulation is to model = 2C ν (S V) ⋅ (S V) εˆ1 t k the solenoidal dissipation by relating it to the enstrophy. A v 1 v traditional modeled version of the turbulent dissipation is − C (εˆ + 4ν∇ ⋅{1 ∇k − ∇ ⋅[ν (S V)]})2 (37) given by εˆ2 k 3 t v v 2 2 ∂ε ε −νν t{Cεˆ3 (∇ V) + ()Cεˆ4 k [()∇k ⋅∇]V}⋅ (∇ V) + V ⋅∇ε = 2Cε1ν t (S V) ⋅ (S V) ∂t k + ∇ ⋅[(ν +ν σ ˆ )∇εˆ] (32) t ε ε 2 − Cε 2 + ∇ ⋅[(ν +ν t σ εˆ )∇ε ] Eq. (37) can be recast using the change of variables from k solenoidal dissipation to enstrophy, and the resulting Through direct analogy with this traditional turbulent formulation is written as dissipation equation, a transport equation for the solenoidal ν = C k ζ 1 2 (38) dissipation can be written t ν v v v v ∂εˆ εˆ v v ∂k + (V ⋅∇)k = 2ν t (S V)⋅ (S V) + V ⋅∇εˆ = 2C ˆ ν (S V)⋅ (S V) ∂t ε1 t k ∂t v (33) v 2 −ν (ζ + 4∇ ⋅{1 ∇k − ∇ ⋅[ν S(V)]}) (39) εˆ 3 t − Cεˆ2 + ∇ ⋅[(ν +ν t σεˆ )∇εˆ] v k + ∇ ⋅((ν +ν σ ){5 ∇k − 2∇ ⋅[ν S(V)]}) t k 3 t Using the relations given in Eq. (28), the solenoidal ∂ζ dissipation transport equation can be recast in terms of the + V ⋅∇ζ enstrophy. This results in the model ∂t v v ζ v v ν = C k ζ 1 2 (34) = 2C ν (S V) ⋅ (S V) t ν ζ 1 t k v ν v ∂k − C (ζ + 4∇ ⋅{1 ∇k − ∇ ⋅[ν (S V)]})2 (40) + (V ⋅∇)k ζ 2 3 t ∂t k v v v 2 2 1 −ν t{Cζ 3 (∇ V) + (Cζ 4 k)[(∇k) ⋅∇]V}⋅ (∇ V) = 2ν t (S V)⋅ (S V) −ν (ζ + 4∇ ⋅{ ∇k −∇ ⋅[ν tS(V)]}) (35) 3 v v + ∇ ⋅[(ν +ν t σ ζ )∇ζ ] + ∇ ⋅((ν +ν σ ){ 5 ∇k −2∇ ⋅[ν S(V)]}) t k 3 t where Cν, σk, Cζ1, Cζ2, Cζ3, Cζ4, and σζ are the closure constants for the model and need to be evaluated. The closure coefficients Cζ1, Cζ2, Cζ3, Cζ4, and σζ could possibly be taken from those given in the study. 7 4. General Enstrophy Closure isolated characteristics of turbulent flow. The model developer is free to choose the actual flows used to evaluate A close look at closure methods 1 through 3 suggests a the closure coefficients. However, the resulting model will general model which would encompass the three models. be tuned to produce more accurate solutions for flows that This formulation is written are similar to those flows used to calculate the closing 1 2 coefficients. ν t = Cν k ζ (41)
∂k IV. CONCLUSION + (V ⋅∇)k ∂t v v v The focus of the proposed research is to evaluate = 2ν (S V)⋅ (S V) −ν (ζ + 4∇ ⋅{1 ∇k − ∇ ⋅[ν S(V)]}) (42) methods to closing the Phillips vorticity-based turbulence t 3 t v model. The completion of this project will in no way be + ∇ ⋅((ν +ν σ ){5 ∇k − 2∇ ⋅[ν S(V)]}) t k 3 t dependent on the success or failure of the new turbulence model to provide a more accurate approach to modeling ∂ζ + V ⋅∇ζ turbulence. The proposed research is an evaluation of the ∂t Phillips vorticity-based turbulence model, not a validation . v v ζ v v = C ν (S V) ⋅ (S V) − C ζ 3 2 1 t k 2 REFERENCES v ν 1 2 Baldwin, B. S., and Barth, T. J. (1990), “A One-Equation − C3 (ζ + C4∇ ⋅{ ∇k − ∇ ⋅[ν t (S V)]}) (43) k 3 Turbulence Transport Model for High Reynolds Number 2 2 Wall-Bounded Flows,” NASA TM–102847. −ν t{C5 (∇ V) + (C6 k)[(∇k) ⋅∇]V}⋅ (∇ V)
+ ∇ ⋅[(ν +ν σ )∇ζ ] − C (ν +ν σ )∇ζ ⋅∇ζ ζ t ζ 7 t ζ Bernard, P. S., and Wallace J. M., (2002a), “Turbulent Flow Analyses,” Turbulent Flow , Wiley, New Jersey, pp. 14–16. where Cν, σk, σζ, and C1 through C7 are the closure constants for the model and need to be evaluated. The proposed research will concentrate on evaluating Bernard, P. S., and Wallace J. M., (2002b), “Averaging and the closure coefficients for the General Enstrophy Closure Turbulent Fields,” Turbulent Flow , Wiley, New Jersey, pp. Model given above. The validity of the first three closure 6–7. methods will then be assessed as they are special cases of the General Enstrophy Closure Model with some pre- Boyer, C. B., and Merzback, U. C., (1989), A History of determined coefficients. It is not yet apparent which closure Mathematics , 2nd ed., Wiley, New York, pp. 656–762. methods will be the most promising. Bradshaw, P., (1994), Experiments in Fluids, 16, 203 C. Closure Coefficient Evaluation Closure coefficients for the resulting turbulence model Frisch, U., and Orszag, S., (1990), Phys. Today . 43, 32. can be obtained several ways. If the closing equations for the model are taken from existing closing equations, the Goldstein, S., (1969) Quotes Sir Horace Lamb of 1932, existing closure coefficients can be used as an initial guess ARFM, 1, 23. for the closure coefficients of the resulting model. However, it cannot be expected that these closure coefficients will be Hamilton, W. R., (1844), “On Quaternions: Or a New entirely viable for the new model presented here because the System of Imaginaries in Algebra,” Philosophical Phillips vorticity-based turbulence model is founded on a Magazine , Vol. 25, No. 3, pp. 489–495. length scale and turbulent-energy equation which differ from the traditional length scale and turbulent-energy Hamilton, W. R., (1853), Lectures on Quaternions, Hodges equation. Therefore, most closure coefficients in the and Smith, Dublin, pp. 1–736. proposed turbulence models will need to be evaluated. Analytical methods are often used to evaluate closure Hamilton, W. R., (1866), Elements of Quaternions , coefficients. In the ideal situation, the coefficients could be Longmans, Green and Co., London, pp. 1–762. evaluated by isolating each term of the equations and comparing the isolated terms to physical aspects of the flow. Hinze, J. O., (1975), “General Introduction and Concepts,” The terms in the governing equations do not appear in a Turbulence , Second Edition, McGraw-Hill, New York, pp. completely isolated form in nature, so empirical relations 1–2. are often developed by employing wind tunnel tests, DNS simulations, or other computational methods that simulate 8 Holmes P. J., Berkooz G., and Lumley J. L., (1996), Turbulence, coherent structures, dynamical systems and Wilcox D. C., (2006a), “Direct Numerical Simulation,” symmetry , Cambridge University Press. Turbulence Modeling for CFD , Third Edition, DCW Industries, Inc., La Cañada, California, pp. 431–435. Jones, W. P., and Launder, B. E., (1972), "The Prediction of Laminarization with a Two-Equation Model of Turbulence,” Wilcox D. C., (2006b), “Reynolds Averaging,” Turbulence International Journal of Heat and Mass Transfer , Vol. 15, Modeling for CFD , Third Edition, DCW Industries, Inc., La pp. 301–314. Cañada, California, pp. 34–38.
Kolmogorov, A. N., (1942), “Equations of Turbulent Wilcox D. C., (2006c), “The Turbulence Energy Equation,” Motion of an Incompressible Fluid,” Izvestia Academy of Turbulence Modeling for CFD , Third Edition, DCW Sciences, USSR; Physics , Vol. 6, Nos. 1 and 2, pp. 56–58. Industries, Inc., La Cañada, California, pp. 108–111.
Kreuzinger, J., Friedrich, R., and Gatski, T. B., (2006), Wilcox D. C., (2006d), “Two-Equation Models,” “Compressibility Effects in the Solenoidal Dissipation Rate Turbulence Modeling for CFD , Third Edition, DCW Equation: A Priori Assessment and Modeling,” Industries, Inc., La Cañada, California, pp. 122–133. International Journal of Heat and Fluid Flow , Vol. 27, pp. 696–706. Zeierman, S., and Wolfshtein, M., (1986), “Turbulent Time Scale for Turbulent Flow Calculations,” AIAA Journal , Vol. Moin, P., and Mahesh, K., (1998), “Direct Numerical 24, No. 10, pp. 1606–1610. Simulation – A Tool in Turbulence Research,” Annual Review of Fluid Mechanics , Vol. 30, pp. 539–578.
Navier, C. L. M. H., (1823), “M émoire sur les Loisdu Movement des Fluides,” Mém. de l’ Acad. R. Sci. Paris , Vol. 6, pp. 389–416.
Phillips, W. F., (2008a), “Turbulent Fluctuations and Reynolds Averaging,” Aerodynamics of Flight DRAFT, Wiley, Hoboken, New Jersey, pp. 8.
Phillips, W. F., (2008b), “Turbulent Fluctuations and Reynolds Averaging,” Aerodynamics of Flight DRAFT, Wiley, Hoboken, New Jersey, pp. 1–10.
Phillips, W. F., (2008c), “Turbulent Flow in Newtonian Fluids,” Aerodynamics of Flight DRAFT, Wiley, Hoboken, New Jersey, pp. 1–51.
Phillips, W. F., (2008d), “An Alternate Turbulent Frequency and Length Scale,” Aerodynamics of Flight DRAFT, Wiley, Hoboken, New Jersey, pp. 48–51.
Phillips, W. F., (2008e), “Characteristic Turbulence Velocity and Length Scales,” Aerodynamics of Flight DRAFT, Wiley, Hoboken, New Jersey, pp. 41–47.
Spalart, P. R., and Allmaras, S. R., (1992), “A One- Equation Turbulence Model for Aerodynamic Flows,” AIAA Paper 92–439.
Stokes, G. G., (1845), “On the Theories of Internal Friction of Fluids in Motion,” Transactions of the Cambridge Philosophical Society , Vol. 8, pp. 287–305.