当前位置：文档之家 › Wall Function Boundary Conditions Including Heat Transfer and Compressibility

Wall Function Boundary Conditions Including Heat Transfer and Compressibility

  • 格式：pdf
  • 大小：1.95 MB
  • 文档页数：8

AIAA J OURNALV ol.42,No.6,June 2004Wall Function Boundary Conditions IncludingHeat Transfer and CompressibilityR.H.Nichols ∗University of Alabama in Birmingham,Birmingham,Alabama 35294andC.C.Nelson †Jacobs Sverdrup Arnold Engineering Development Center Group,Arnold Air Force Base,Tennessee 37389Wall function boundary conditions that include the effects of compressibility and heat transfer are developed for transport-type turbulence models.The wall functions are based on coupled velocity and temperature profiles valid in both the viscous sublayer and the log layer.The wall function boundary conditions are developed within a rigid set of assumptions so that a consistent set of relationships are derived for the wall shear stress,the wall heat transfer,and the turbulence quantities at the first point off the wall.The new boundary conditions are incorporated into a Navier–Stokes computational fluid dynamics code that includes the Spalart–Allmaras and shear stress transport turbulence models.The wall function boundary conditions produce reasonable engineering solutions for the test cases presented for initial wall spacings of y +<–100.NomenclatureB=law-of-the-wall constant C f =skin friction C h =Stanton number C p =pressure coefficientC µ=two-equation turbulence model eddy-viscosity constant C ν1=Spalart–Allmaras turbulence model constant K =molecular thermal conductivity K t =turbulent thermal conductivity k =turbulent kinetic energy M =Mach number P =pressurePr =molecular Prandtl number P t =total pressure q w =wall heat transfer Re =Reynolds numberRe x =Reynolds based on the distance from the leading edge of a plater =recovery factor,Pr 1/3T =temperatureT t =total temperature T w =wall temperature u =velocity parallel to the wall u τ=friction velocity,√(τw /ρw )y =coordinate direction normal to the wall ε=turbulent dissipation κ=law-of-the-wall constant µ=molecular viscosity µt =eddy viscosityµw =wall molecular viscosity ρ=density ρw =wall density τw =wall shear stressω=specific turbulent dissipationReceived 2July 2003;revision received 29January 2004;accepted for publication 10February 2004.This material is declared a work of the ernment and is not subject to copyright protection in the United States.Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00per-copy fee to the Copyright Clearance Center,Inc.,222Rosewood Drive,Danvers,MA 01923;include the code 0001-1452/04$10.00in correspondence with the CCC.∗Research Associate Professor,Department of Mechanical Engineering.Senior Member AIAA.†Senior Engineer.Senior Member AIAA.IntroductionTURBULENT transport models that are applicable all of the way to the wall are called low-Reynolds-number turbulence models because they are valid at low turbulent Reynolds numbers [ρk 2/(µε)].Low-Reynolds-number models require tight grid spac-ing near the wall to resolve the large gradients in velocity,tem-perature,and turbulent quantities near the wall.This can lead to large grid requirements for three-dimensional Navier–Stokes ap-plications.The stability of a numerical algorithm is limited by the smallest cell size in a grid,so that the small cells near the wall can severely restrict the maximum allowable time step for a prob-lem.The near-wall turbulence damping terms are often expressed in terms of exponential functions.This can lead to difficulties in converging the numerical solution in the near-wall region.Turbulence models that do not include wall correction terms in the differential equations are called high-Reynolds-number turbu-lence models.High-Reynolds-number turbulence models rely on empirically derived algebraic models of the near-wall region of the boundary layer to provide boundary condition information to the mean flow Navier–Stokes equations at the first point off the wall.These models of the near-wall region are called wall functions.Be-cause the high-gradient region near the wall is modeled with these empirical relationships,the first point off the wall may be placed much farther away than with low-Reynolds-number models.This reduces the number of points required to discretize a flowfield and increases the maximum allowable time step.Both of these advan-tages are especially useful in modeling unsteady viscous flows.Most of the early wall function boundary conditions were based on the analysis of Millikan 1and Tennekes and Lumley 2for chan-nels and flat plates.Most of the early models did not include the effects of pressure gradient or heat transfer.Nichols,3Viegas et al.,4and Smith 5have all developed wall functions that included heat transfer at the wall.These formulations were not based on a single unified velocity profile,so that the wall functions were often discon-tinuous in the transition region from the sublayer to the log layer.Wilcox 6introduced a wall-matching methodology in which a set of differential equations were used to integrate from the wall to the first point off the wall.The wall-matching strategy requires a differ-ent set of differential equations for each different turbulence model.Frink 7developed a set of wall functions for adiabatic walls based on the unified theory of Spalding.8Shih et al.9describe a unified wall function also based on Spalding’s work,which includes pressure gradient effects but does not include coupling with heat transfer at the wall.This paper describes a wall function boundary condition based on coupled velocity and temperature boundary-layer profiles1107D o w n l o a d e d b y "U n i v e r s i t y o f M i c h i g a n , M e d i a U n i o n L i b r a r y " o n J a n u a r y 8, 2013 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.35391108NICHOLS AND NELSONthat are applicable to incompressible flows,compressible flows,and flows with heat transfer.The wall function boundary conditions are developed within a rigid set of assumptions so that a consistent set of relationships are derived for the wall shear stress,wall heat trans-fer,and the turbulence quantities at the first point off the wall.This paper also describes the methodology for implementing the wall function boundary condition into a Navier–Stokes flow solver.The wall functions described here are compatible with most transport-type turbulence models.TheoryThere are six fundamental assumptions used in the development of wall function boundary conditions for compressible flows:1)Analytical expressions are available for the velocity and tem-perature profiles in the lower part of the boundary layer.2)Analytical expressions are available for the turbulent transport variables at the first point off the wall.3)There is no pressure gradient normal to the wall in the lower part of the boundary layer.4)The shear stress,τw =τ=(µ+µt )(∂u /∂y ),is constant in the lower part of the boundary layer.5)The heat transfer,q w =q =(K +K t )(∂T /∂y ),is constant in the lower part of the boundary layer.6)There are no chemical reactions,that is,the chemistry is frozen,in the lower part of the boundary layer.Several empirical relationships are available for the velocity in a boundary layer that can be used to develop wall function boundary conditions.A typical boundary-layer velocity profile is shown in Fig.1.The simplest form is the adiabatic incompressible universal law-of-the-wallu +=(1/κ)log (y +)+B(1)where u +and y +are defined asu +=u /u τ(2)y +=ρw u τy /µw(3)The universal law-of-the-wall is valid for the y +range of about 30<y +<1000.The velocity in the sublayer,0<y +<10,is given byu +=y +(4)The sublayer must be blended with the law-of-the-wall to cover the interim region.Spalding 8suggested a unified form valid for the log layer and the sublayer as well as the transition region.This form is given byy +=u ++e −κBe κu +−1−κu +−(κu +)2/2−(κu +)3/6(5)Note that the e −κB e κu +term in Eq.(5)is a restatement of the in-compressible adiabatic law-of-the-wall [Eq.(1)].The constants κand B are generally taken as 0.4and 5.5,respectively.The wall function boundary condition can be tuned for such effects as sur-face roughness by adjusting the κand B pressibility and heat transfer effects can be included by replacing the incom-pressible adiabatic law-of-the-wall term in Spalding’s equation(5)Fig.1Typical boundary layer profile.with the outer velocity form of White and Christoph.10White and Christoph’s outer velocity is given byu +=(1/2 ) β+Q sin + √ κ ln y + y +0(6)where =r u 2τ2c p T w,β=q w µwρw T w k w u τ,ϕ=sin −1−βQQ =(β2+4 )12,y +0=e −κB(7)The nondimensional parameter models compressibility effects,and the parameter βmodels heat transfer effects.The wall shear stress τw appears in the definitions of u +,y +, ,and β.The wall heat transfer q w appears only in the definition of β.The original formulation of the outer velocity also included a parameter for pres-sure gradient in Eq.(10).The pressure gradient parameter has been ignored in this effort because of the difficulty of determining a pres-sure gradient parameter for general three-dimensional flows.The outer velocity form of White and Christoph reduces to the standard law-of-the-wall [Eq.(1)]for incompressible adiabatic flow.Equa-tion (5)can then be written asy +=u ++y +White −e −κB [1+κu ++(κu +)2/2+(κu +)3/6](8)wherey +White=exp κ √{sin −1[(2 u +−β)/Q ]−φ}e −κB(9)The temperature distribution within the boundary layer is given bythe Crocco–Busemann equation 10T =T w [1+βu +− (u +)2](10)Hence,the boundary-layer profile is defined in terms of the shear stress τw and the wall heat transfer q w .For adiabatic wall cases,q w =0,and the Crocco–Busemann equation reduces toT =T w {1+[(γ−1)r /2]u 2}(11)White and Christoph validated the ability of their outer law to predict skin friction with experimental data for 426adiabatic wall cases and with 147cold-wall cases in Ref.10and found that the absolute error [(1/N )|e i |]was 5.17%for the adiabatic cases and 12.29%for the cold wall cases.The absolute error of their correlation was about the same as the best boundary-layer correlations of the time.The wall shear stress for adiabatic flows is determined as follows:1)Set the velocity at the wall to zero for nonmoving body prob-lems or to the grid velocity for moving body problems.2)Given the velocity and temperature at the first point off the wall;solve Eq.(11)for the wall temperature T w .3)Extrapolate the pressure from the first point off the wall and solve for the wall density ρw using the equation of state.4)Iteratively solve Eq.(8)for the wall shear stress τw in the wall coordinate system given the distance of the first point off the wall from the wall.5)Rotate the stress tensor into the computational coordinate system,and replace the viscous flux calculation at the wall for cell-centered algorithms or at the half-node for node-centered algorithms.The wall shear stress and the heat transfer for constant temperature wall flows are determined as follows:1)Set the velocity at the wall to zero for nonmoving body prob-lems or to the grid velocity for moving body problems.2)Extrapolate the pressure from the first point off the wall and solve for the wall density using the equation of state with the given wall temperature.3)Iteratively solve Eqs.(8)and (10)for the wall shear stress τw and heat transfer q w in the wall coordinate system given the distance of the first point off the wall from the wall.D o w n l o a d e d b y "U n i v e r s i t y o f M i c h i g a n , M e d i a U n i o n L i b r a r y " o n J a n u a r y 8, 2013 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.3539NICHOLS AND NELSON 11094)Rotate the stress tensor and the heat transfer vector into the computational coordinate system,and replace the viscous flux cal-culation at the wall for cell-centered algorithms or at the half-node for node-centered algorithms.One approach to introducing the wall function corrected wall shear stress and heat transfer into the calculation of the viscous fluxes has been to calculate an effective wall eddy viscosity so that the discrete shear stress at the first-half point off the wall given by τ1.5=[(µ1+µ2)/2+(µt 1+µt 2)/2][(u 2−u 1)/(y 2−y 1)](12)yields the correct value for the wall shear stress.Unfortunately,this approach introduces errors into the energy equation because the slope of the temperature derivative will not be calculated correctly,that is,a separate effective wall eddy viscosity would be required for the temperature.Sondak and Pletcher 11describe a procedure to perform a trans-formation of the stresses for generalized coordinate systems.Al-though this transformation is much more general than the effective wall eddy-viscosity approach and introduces no error in the energy equation,it is also much more complex.In their transformation,the wall function-generated wall shear stress is assumed to be in the wall coordinate system,that is,a system aligned with velocity vector at the first point off the wall and the normal vector to the wall.This is not always the case for separated flows,but the assumption seems to produce reasonable results even when the flow is separated.Another method of correcting the computational shear stress at the wall with the wall function value is to calculate the ratio of the mag-nitude of the computed wall shear stress to the wall function value and then scale the computed values by this ratio.This is equivalent to performing the transformation of Sondak and Pletcher 11and is much simpler to apply.The temperature derivative at the wall may be treated in a similar manner.Once the wall shear stress has been calculated,the turbulence transport variables must be determined at the first point off the wall.The eddy viscosity at the first point off the wall for incompressible adiabatic flow can be found using the constant stress assumption to beµt /µw =κe −κBe κu +−1−κu +−(κu +)2/2(13)The equivalent form of the eddy viscosity for compressible flowswith heat transfer can also be found using the constant stress as-sumption.The eddy viscosity is given byµt µw =1+∂y +White ∂y +−κe −κB 1+κu ++(κu +)22−µw +1µw(14)where µw +1is the molecular viscosity at the first point off the walland ∂y +White /∂y +is given by∂y +White ∂y +=2y +White κ√ Q 1−(2 u +−β)2Q 212(15)Equation (14)reduces to Eq.(13)for adiabatic incompressibleflows.This eddy-viscosity formulation provides a Wilcox 6-like wall matching for the turbulence variables if the eddy viscosity is used to define the turbulence variable at the first point off the wall.This eddy-viscosity formulation is independent of the turbulence model chosen.Turbulence Model Boundary ConditionsThe values of the transport model turbulence variables at the first point off the wall must also be defined.The values for the Spalart–Allmaras 12variable ˜νcan be found by iteratively solving the following equation:˜ν4=µt ˜ν3+µt C 3v 1(16)The turbulent kinetic energy and turbulent dissipation at the first point off the wall for k –εturbulence models are given byk =u 2τC µ(17)ε=C µρk 2µt(18)The turbulent kinetic energy and specific turbulent dissipation atthe first point off the wall for k –ωand shear stress transport (SST)turbulence models are given byωi =6µw 0.075ρw y 2(19) o =u τµκy (20) =2i + 2o(21)k =µt(22)The two-level model for ωin Eqs.(19–22)was suggested by Veiser et al.13Additional Modifications to Flow Solvers for Wall FunctionsMost upwind flow solvers require no further modifications than the inclusion of the corrected wall stress and heat transfer in the viscous flux calculations.The smoothers used with central differ-ence algorithms require some additional modifications.The fourth-order smoother used to maintain numerical stability will produce excessive dissipation at the wall when wall functions are employed because of the large velocity difference between the wall and the first point off the wall.To prevent this from occurring,the smoother should be modified as follows.The standard form of the explicit fourth-order smoother in the ξcomputational direction ist ∇ξ ψ(4) ξ ξ(23)whereξ=∇ξ ξ(J q )n +1j(24)where t is the time step, ξis the central difference operator,and ψ(4)is the explicit fourth-order smoothing coefficient.The second derivative term ξis normally set to zero at the wall for viscous calculations.This term is also set to zero at the first point off the wall when wall functions are used.This effectively turns the fourth-order smoothing off at the first point off the wall and reduces it significantly at the second point off the wall.Once the wall shear stress is calculated using wall functions,the y +value of the first point off the wall is known.It is a simple matter to switch automatically between the wall functions and to integrate to the wall based on the local value of y +.The turbulence model chosen should contain the low-Reynolds-number terms required for the model to be valid in the near-wall region if automatic switching is implemented.Automatic switching offers advantages in complex configurations because computational grid points can be saved by using the larger wall function wall spacings on nonessential portions of the geometry and integrating to the wall where high-quality skin friction or heat transfer is required.Automatic switching is also useful in grid sequencing or multigrid algorithms where the wall functions can provide an improved estimate of the wall shear stresses and heat transfer on the coarse mesh solutions.ApplicationsThe wall function boundary conditions described earlier were incorporated into the NXAIR 14three-dimensional overset Navier–Stokes code.NXAIR is an implicit upwind Navier–Stokes code that employs a Newton subiteration scheme.The code is second-order accurate in time and third-order accurate in space.NXAIR includes several implicit solution algorithms and allows for gridD o w n l o a d e d b y "U n i v e r s i t y o f M i c h i g a n , M e d i a U n i o n L i b r a r y " o n J a n u a r y 8, 2013 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.35391110NICHOLS ANDNELSONFig.2Flat plate skin friction predictions for Spalart–Allmaras tur-bulence model with wall functions for varying initial wall grid pointspacing.Fig.3Flat plate skin-friction predictions for the SST turbulence model with wall functions for varying initial wall grid point spacing.sequencing and multigrid solution cycles.The turbulence models are solved decoupled from the mean flow equations at each Newton subiteration.The Spalart–Allmaras one-equation and the SST two-equation turbulence models contained in NXAIR were modified to include the wall function boundary conditions.The wall function boundary condition was applied explicitly,although the Newton subiteration tends to give an implicit flavor to the boundary condition implementation.The code uses automatic switching when the wall function option is selected.The wall functions currently perform the automatic switching at y +=5.Grid Sensitivity for a Flat Plate with Adiabatic WallsThe predicted and theoretical values for the skin-friction distribu-tion on an adiabatic flat plate with varying initial grid wall spacings using the Spalart–Allmaras turbulence model are shown in Fig.2and using the SST turbulence model are shown in Fig.3.The theoret-ical values are from White.15The y +=1results were not performed with wall functions and are included for reference.All of the results shown here used a grid stretching ratio of 1.2.The wall function pre-dictions of skin friction are very similar for both turbulence models.The models are in reasonable agreement length Reynolds numbers Re x above 1.0×106.Velocity profiles for the two turbulence mod-els are shown in Figs.4and 5.Again the y +=1.0results were not run with wall functions and are included for reference.The velocity profiles are in reasonable agreement with each other and with theory for all of the wall spacings run here.Grid Sensitivity for an Axisymmetric BumpA second example of the grid sensitivity of the wall function boundary conditions for adiabatic walls that includes a pressure gradient is the NASA Ames Research Center transonicaxisymmet-Fig.4Flat plate velocity profile predictions for Spalart–Allmaras tur-bulence model with wall functions for varying initial wall grid pointspacing.Fig.5Flat plate velocity profile predictions for SST turbulence model with wall functions for varying initial wall grid pointspacing.Fig.6Geometry for the NASA Ames Research Center axisymmetric bump experiment.ric bump experiment.16The geometry is shown in Fig.6.The model consisted of a sharp-lipped hollow cylinder with a 15.2-cm outer surface diameter.The bump was a circular arc 20.3cm long and 1.9cm high that begins 60.3cm downstream of the cylinder leading edge.The upstream intersection of the bump and cylinder was faired with a circular arc.The test was run at a Mach number of 0.875and a chord Reynolds number of 2.67×106.Surface pressure and laser velocimetery data were obtained on this configuration.Both the Spalart–Allmaras and the SST models were run with varying initial grid wall spacings.Results for the pressure coefficient for the Spalart–Allmaras model and the SST model are shown in Figs.7and 8,respectively.Results for the velocity distribution at x /c =1.0(the trailing edge of the bump)are shown for the Spalart–Allmaras and the SST models in Figs.9and 10,respectively.The y +=1.0results were not run with wall functions and are included for reference.For both the pressure coefficient distribution and the velocity profile at x /c =1.0,the results for each model are similar.D o w n l o a d e d b y "U n i v e r s i t y o f M i c h i g a n , M e d i a U n i o n L i b r a r y " o n J a n u a r y 8, 2013 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.3539NICHOLS AND NELSON1111Fig.7Pressure coefficient predictions for NASA Ames Research Cen-ter axisymmetric bump for Spalart–Allmaras turbulence model with wall functions for varying initial wall grid pointspacing.Fig.8Pressure coefficient predictions for NASA Ames Research Cen-ter axisymmetric bump for SST turbulence model with wall functions for varying initial wall grid pointspacing.Fig.9Velocity distribution at x /c =1.0for NASA Ames Research Cen-ter axisymmetric bump for Spalart–Allmaras turbulence model with wall functions for varying initial wall grid point spacing.The results tend to start to diverge for y +=200but are in good agreement for y +values of 100or less.Note that the wall function produce reasonable results for this shock-induced boundary-layer separation.Grid Sensitivity for a Flat Plate with Heat TransferCalculating heat transfer accurately can be more difficult than predicting skin friction.This can be seen in the subsonic flat plate example when the wall temperature is specified to be 1.5times the freestream temperature.The sensitivity of the skin friction andheatFig.10Velocity distribution at x /c =1.0for NASA Ames Research Center axisymmetric bump for SST turbulence model with wall func-tions for varying initial wall grid pointspacing.Fig.11Effect of wall spacing on skin friction on a flat plate with heat transfer using Spalart–Allmaras turbulencemodel.Fig.12Effect of wall spacing on the heat transfer (Stanton number)on a flat plate using Spalart–Allmaras turbulence model.transfer result with varying initial grid wall spacing is shown in Figs.11and 12for the Spalart–Allmaras model and in Figs.13and 14for the SST model.The theoretical results are from White.15The y +=0.1results were not run with wall functions and are included for comparison.The grid stretching ratio was fixed at 1.2for these results.Both the skin friction and heat transfer seem to be relatively insensitive to the wall spacing for the wall spacings presented here above a length Reynolds number Re x =1×106.Profiles of velocity and temperature at a length Reynolds number Re x =1×107are shown in Figs.15and 16,respectively,for the Spalart–Allmaras model and in Figs.17and 18for the SST model.Again the y +=0.1results were not run with wall functions andD o w n l o a d e d b y "U n i v e r s i t y o f M i c h i g a n , M e d i a U n i o n L i b r a r y " o n J a n u a r y 8, 2013 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.35391112NICHOLS ANDNELSONFig.13Effect of wall spacing on skin friction on a flat plate with heat transfer using SST turbulencemodel.Fig.14Effect of wall spacing on heat transfer (Stanton number)on a flat plate using SST turbulencemodel.Fig.15Effect of wall spacing on velocity profile on a flat plate with heat transfer using Spalart–Allmaras turbulence model and wall functions.are included for comparison purposes.The wall function results are in reasonable agreement for all wall spacings investigated for both turbulence models.Grid Sensitivity for a Nozzle with Heat TransferFlow through a supersonic nozzle with a constant temperature wall can serve as a test case for evaluating the performance of the turbulence model in the presence of strong pressure gradients.Back et al.17measured the wall pressure distribution and heat transfer for a converging–diverging nozzle with a throat diameter of 0.0458m and an exit diameter of 0.1227m.High-pressure air was heated by the internal combustion of methanol and flowed along a cooled constant area duct with a length of 0.4572m and a diameter of 0.355m before entering the nozzle.The nozzle geometry and boundaryconditionsFig.16Effect of wall spacing on temperature profile on a flat plate with heat transfer using Spalart–Allmaras turbulence model and wallfunctions.Fig.17Effect of wall spacing on velocity profile on a flat plate with heat transfer using the SST turbulence model and wallfunctions.Fig.18Effect of wall spacing on temperature profile on a flat plate with heat transfer using the SST turbulence model and wall functions.are shown in Fig.19.The gas could be treated as a calorically perfect gas with a ratio-of-specific heats γof 1.345.The nozzle exit Mach number was approximately 2.5.The molecular viscosity and thermal conductivity were assumed to vary according to Sutherland’s law.The grid-stretching ratio used in this study was parisons of the predicted and measured pressure along the nozzle are shown in Fig.20for the Spalart–Allmaras model and in Fig.21for the SST model.The results for a wall spacing of y +=0.1were not run with wall functions and are included for reference.The pressure distribution is seen to be insensitive to the wall spacing.The wall heat transfer is shown in Fig.22for the Spalart–Allmaras model and in Fig.23for the SST model.The SST results for heat transfer areD o w n l o a d e d b y "U n i v e r s i t y o f M i c h i g a n , M e d i a U n i o n L i b r a r y " o n J a n u a r y 8, 2013 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.3539NICHOLS AND NELSON1113Fig.19Nozzlegeometry.Fig.20Effect of wall spacing on the pressure distribution for a su-personic nozzle with heat transfer using Spalart–Allmaras turbulence model and wallfunctions.Fig.21Effect of wall spacing on pressure distribution for a super-sonic nozzle with heat transfer using SST turbulence model and wallfunctions.Fig.22Effect of wall spacing on heat transfer distribution for a su-personic nozzle with heat transfer using Spalart–Allmaras turbulence model and wallfunctions.Fig.23Effect of wall spacing on heat transfer distribution for a su-personic nozzle with heat transfer using SST turbulence model and wall functions.much less sensitive to the initial wall spacing than are the Spalart–Allmaras results.The heat transfer predictions are adequate for many applications even for an initial wall spacing of y +=100and are probably within the accuracy of the correlations used to define the wall function.ConclusionsA general set of wall function boundary conditions has been de-veloped and demonstrated for compressible flows with and without heat transfer at the wall.The wall functions are based on coupled pro-files for velocity and temperature in the lower part of the boundary layer.The wall function boundary conditions are developed within a rigid set of assumptions so that a consistent set of relationships are developed for the wall shear stress,wall heat transfer,and the turbulence quantities at the first point off the wall.This results in wall functions that are valid in the sublayer,in the log layer,and in the transition region between these regions.The wall functions can be automatically switched on or off locally based on the local value of y +when they are used with in conjunction with a low-Reynolds-number turbulence model formulation.The wall function boundary conditions were incorporated into two transport-type turbulence models.The computed results with the wall functions were in good agreement with theory and with low Reynolds number results for all of the cases attempted here for wall spacings of y +≤100.These larger wall spacings significantly reduce the number of points required to obtain a turbulent Navier–Stokes solution and significantly reduce the maximum Courant–Friedrichs–Lewy number for a viscous solution.AcknowledgmentsThe research reported herein was performed by the Arnold Engineering Development Center (AEDC),Air Force Materiel Command.Work and analysis for this research were performed by personnel of the University of Alabama at Birmingham and by personnel of Jacobs Sverdrup AEDC Group,technical services con-tractor for AEDC.D o w n l o a d e d b y "U n i v e r s i t y o f M i c h i g a n , M e d i a U n i o n L i b r a r y " o n J a n u a r y 8, 2013 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.3539。

合集下载

OpenFoam标准应用

OpenFoam标准应用

wallShearStress yPlusLES yPlusRAS 紊流后处理
细化表面附近单元 去除短边，合并共线边 查找同一单元上由多个面域组成的面，并进行合并。多面域组成的面会导致一 些错误，如删除细化的相邻单元，剩下的 4 个外部面有相同的属主 处理网格元素 PDR 类型的分析，对场和网格进行预准备 细化六面体网格，单元拆分为 2x2x2 计算笛卡尔网格的细化水平，贴合之前运行。 细化靠近面域的单元 去除面（在两面合并单元） 选择与表面有关的单元 拆分有平面的单元
多块网格生成器 从存在的面域挤出网格（默认面的法向向外，可以翻转面），也可以从文件读取 面域 使用 2D 网格（每面只有两个点，没有前后面），通过挤出指定的厚度生成 3D 网格。 将面区域挤出为分离的网格，例如为创建液膜区域 保形泰森多边形法自动网格生成器 编写背景网格，由 foamyHexMesh 和结构 distanceSurface 创建
使用规定的网格修改器连接拓扑分离的网格 根据用户指定的特征角，将外部面分为面域 监察网格有效性 将内部面作为边界面，不创建点，与 mergeOrSplitBaffles 不同 创建未选择边界面的面域。面来自存在的面域或 faceSet 使用位移场和标量因数使 polyMesh 变形 削平 2D 笛卡尔网格的前后面 接收单元中心位于单元表面的单元，要求表面闭合、单通。 合并网格 查找共用点的面，合并面或复制点 镜像网格 网格运动，网格拓扑改变 移动网格，发动机计算 移动网格 读取 obj 线，转化为 vtk 修正面区域的方向 计算 polyMesh 的对偶 依附所有特征和面域边缘 多方向细化单元 给单元重新编号以减小频宽，从所有时间目录中读取场并重新编号 交互式处理单元、面、点的系列、或区域 从 pointSets/faceSets/cellSets 中,给网格增加 pointZones/faceZones/cellZone 读 取 场 并 映 射 到 网 格 ， 去 除 所 有 内 部 面 (singleCellFvMesh) ， 写 入 区 域 singleMesh.。目的是生成只用于边界的网格和场。可能很容易就生成非法网格， 只能在 paraFoam 中查看。 使用 attachDetach 将内部面变为外部面，从而拆分网格 将网格拆分为多区域 缝合网格 根据 cellSet 选择一部分网格 通过一个词库，对 cellSets/faceSets/pointSets 进行操作 根据转化、旋转、缩放选项转化网格的点 读取悬挂顶点的网格，zips up 单元以保证多面体单元形状有效，封闭。
流体力学常用名词术语中英文对照

流体力学常用名词术语中英文对照

流体力学常用名词术语中英文对照CFL条件 Courant- Friedrichs- Lewy condition ,CFL condition KDU方程 KDV equationU形管 U-tubeA埃克特数 Eckert number奥尔-索末菲方程 Orr-Sommerfeld equation奥辛流 Oseen flowB巴塞特力 Basset force板块法 panel method半隐格式 semi-implicit scheme薄翼理论 thin-airfoil theory保单调差分格式monotonicity preserving diffe-rence scheme爆轰 detonation爆炸 explosion毕奥-萨伐尔定律 Biot-Savart law壁剪切速度 friction velocity壁剪应力 skin friction, frictional drag壁效应 wall effect边界层 boundary layer边界层方程 boundary layer equation边界层分离 boundary layer separation边界层厚度 boundary layer thickness边界层理论 boundary later theory边界层转捩 boundary layer transition边界元法 boundary element method变分法 variational method标记网格法 marker and cell method, MAC method 表观粘度 apparent viscosity表面波 surface wave表面力 surface force表面张力 surface tension表面张力波 capillary wave波高 wave height波列 wave train波能 wave energy波群 wave group波速 wave speed, wave velocit波阻 wave drag伯格斯方程 Burgers equation伯努利定理 Bernonlli theorem伯努利方程 Bernoulli equation泊 poise泊肃叶流 Poiseuille flow不定常法 time-dependent method不规则波 irregular wave不可压缩流[动] incompressible flow不可压缩流体 incompressible fluid不可压缩性 incompressibility不稳定性 instability布拉休斯解 Blasius solution部分相似 partial similarityC测速法 anemometry测压孔 pressure tap层流 laminar flow层流边界层 laminar boundary layer层流分离 laminar separation掺气流 aerated flow超空化流 supercavitating flow超空泡 supercavity超空泡流 supercavity flow超声速流[动] supersonic flow超压[强] over pressure潮波 tidal wave沉[降堆]积 sedimentation, deposition沉积物 sediment, deposit沉降速度 settling velocity成核 nucleation尺度效应 scale effect冲击波 shock wave出口 exit, outlet出口压力 exit pressure传播 propagation传导 conduction传感器 transducer, sensor传热系数 heat transfer coefficient传质系数 mass transfer coefficient船波 ship wave次层 sublayer猝发过程 bursting processD达朗贝尔佯廖 d'Alembert paradox大涡模拟 large eddy simulation单调差分格式 monotone difference scheme 单相流 single phase flow单组份流 single-component flow当地马赫数 local Mach number等熵流 isentropic flow低速空气动力学 low-speed aerodynamics狄里克雷边界条件 Dirichlet boundary condition 底压 base pressure地面效应 ground effect定理 pi theorem, Buckingham theorem定倾中心 metacenter动力相似[性] dynamic similarity动力粘性 dynamic viscosity动量方程 momentum equation动量厚度 momentum thickness动量交换 momentum transfer动量守恒 conservation of momentum动态响应 dynamic response动态校准 dynamic calibration堵寒效应 blockage effect堵塞 blockage杜福特-弗兰克尔格式 Dufort-Frankel scheme 对流 convection对流传热 convective heat transfer对流扩散方程 convection diffusion equation对流涡胞 convective cell钝体 blunt body钝头体 bluff body多管压强计 multiple manometer多重尺度问题 multiple scale problem多重网格法 multi- grid methodE二次流 secondary flow二维流 two-dimensional flowF反扩散差分格式 anti-diffusion difference scheme 反流 reverse flow反射 reflection反压 back pressure放大矩阵 amplification matrix放大因子 amplification factor非定常流 unsteady flow, non-steady flow非绝热流 diabatic flow非均匀流 nonuniform flow非平衡流[动] non-equilibrium flow非线性波 nonlinear wave非线性不稳定性 nonlinear instability菲克定律 Fick law沸腾 boiling分布 distribution分步法 fractional step method分层流 stratified flow分离点 separation point分离流 separated flow分子扩散 molecular diffusion风洞 wind tunnel风速管 Pitot- static tube冯.诺伊曼条件 von Neumann condition弗劳德数 Froude number浮体 floating body辐射传热 radiative heat transfer附加质量 added mass ,associated mass附面层 boundary layer附体激波 attached shock wave附着点 attachment point附着涡 bound vortex复势 complex potential复速度 complex velocityG伽辽金法 Galerkin method盖斯特纳波 Gerstner wave高度水头 elevating head高分辨率格式 high resolution scheme高速空气动力学 high-speed aerodynamics 戈本诺夫格式 Godunov scheme格拉斯霍夫数 Grashof number构型 configuration孤立子 soliton拐角流 corner flow管流 pipe flow, tube flow规则波 regular wave过水断面 flow cross-sectionH海洋水动力学 marine hydrodynamics亥姆霍兹定理 Helmholtz theorem含沙流 sediment-laden stream含水层 aquifer焓厚度 enthalpy thickness耗散 dissipation洪水波 flood wave后缘 trailing edge滑移速度 slip velocity环量 circulation环流 circulation缓流 subcritical flow缓燃 deflagration回流 back flow汇 sinkJ积分方法 integral method激波 shock wave激波捕捉法 shock-capturing method激波层 shock layer激波管 shock tube激波管风洞 shock tube wind tunnel激波拟合法 shock-fitting method激波阵面 shock front激光多普勒测速计laser Doppler anemometer,laser Doppler velocimeter急变流 rapidly varied flow急流 supercritical flow几何相似 geometric similarity计算流体力学 computational fluid mechanics计算区域 computational domain迹线 path, path line减阻 drag reduction剪切层 shear layer剪切流 shear flow渐变流 gradually varied flow浆体 slurry降水曲线 dropdown curve交错网格 staggered mesh交替方向隐格式alternating direction implicit scheme, ADIscheme介质 medium紧差分格式 compact difference scheme进口 entrance, inlet近场流 near field flow近似因子分解法 approximate factorization method 静[态]校准 static calibration静压管 static [pressure]tube静压头 static head镜象法 image method局域相似 local similarity矩量法 moment method卷筒涡胞 roll cell绝热流 adiabatic flow均匀流 uniform flowK卡门涡街 Karman vortex street开尔文定理 Kelvin theorem可压缩流[动] compressible flow可压缩流体 compressible fluid克兰克-尼科尔森格式 Crank-Nicolson scheme空化 cavitation空化数 cavitation number空泡流 cavity flow空气动力学 aerodynamics空蚀 cavitation damage孔板流量计 orifice meter孔流 orifice flow控制体积 control volume库埃特流 Couette flow库塔-茹可夫斯基条件 Kutta-Zhoukowski condition 跨声速流[动] transonic flow扩散 diffusion扩散段 diffuser扩散率 diffusivity扩散速度 diffusion velocity扩散性 diffusivityL拉克斯等价定理 Lax equivalence theorem拉克斯-温德罗夫格式 Lax-Wendroff scheme拉瓦尔喷管 Laval nozzle来流 incoming flow兰金-于戈尼奥条件 Rankine-Hugoniot condition 雷诺比拟 Reynolds analogy雷诺数 Reynolds number厘泊 centipoise厘沱 centistoke离解 dissociation离散涡 discrete vortex黎曼解算子 Riemann solver连续介质假设 continuous medium hypothesis连续介质力学 mechanics of continuous media涟漪 ripple量热状态方程 caloric equation of state临界雷诺数 critical Reynolds number临界流 critical flow临界热通量 critical heat flux流[动] flow流场 flow field流出边界条件 outflow boundary condition流动参量 flow parameter流动分离 flow separation流动稳定性 flow stability流动显示 flow visualization流管 stream tube流函数 stream function流量 flow rate, flow discharge流量计 flow meter流面 stream surface流入边界条件 inflow boundary condition流速计 anemometer流态 flow regime流体动力学 fluid dynamics流体网格法 fluid in cell method,FLIC method 流体运动学 fluid kinematics流体质点 fluid particle流线 stream lineM马赫波 Mach wave马赫角 Mach angle马赫数 Mach number马赫线 Mach line马赫锥 Mach cone马蹄涡 horseshoe vortex脉线 streak line毛细[管]作用 capillarity弥散 dispersion明槽流 open channel flow磨擦损失 friction loss磨擦因子 friction factor穆曼-科尔格式 Murman-Cole schemeN纳维-斯托克斯方程 Navier-Stokes equation 内流 internal flow能量传递 energy transfer能量法 energy method能量方程 energy equation能量厚度 energy thickness能量守恒 conservation of energy能量输运 energy transport拟序结构 coherent structure粘度测定法 visco[si] metry粘度计 visco[si] meter粘性流[动] viscous flow凝结 condensation牛顿流体 Newtonian fluid浓度 concentration努塞特数 Nusselt numberO欧拉方程 Euler equation欧拉数 Euler number偶极子 doublet, dipoleP排放量 discharge排水 drainage配置方法 collocation method喷管 Nozzle皮托管 pitot tube频率响应 frequency response平面流 plane flow破碎波 breaking wave普朗特-迈耶流 Prandtl-Meyer flow普朗特数 prandtl number普雷斯顿管 preston tube谱方法 spectral methodQ起动涡 starting vortex气动加热 aerodynamic heating气动力 aerodynamic force气动热力学 aerothermodynamics气动噪声 aerodynamic noise气动中心 aerodynamic center气化 gasification气体动力学 gas dynamics气体润滑 gas lubrication前缘涡 leading edge vortex浅水波 shallow water wave强迫对流 forced convection强守恒型 strong conservation form氢泡法 nydrogen bubble method区域分解 domain decomposition全变差下降格式total variation decreasing scheme TVD schemeR扰动 disturbance, perturbation绕射 diffraction热传导 conductive heat transfer热对流 heat convection热交换 heat exchange热量传递 heat transfer热敏电阻 thermistor热膜流速计 hot- film anemometer热线流速计 hot-wire anemometer热状态方程 thermal equation of state 人工压缩 artificial compression人工粘性 artificial viscosity瑞利流 Rayleigh flow瑞利数 Rayleigh number弱解 weak solution弱守恒型 weak conservation formS三维流 three-dimensional flow散度型 divergence form散射 scattering色散 dispersion熵函数 entropy function熵条件 entropy condition熵通量 entropy flux射流 jet深水波 deep water wave失速 stall施密特数 schmidt number施特鲁哈尔数 Strouhal number时间分步法 time splitting method时间线 time line示踪物 tracer势 potential势流 potential flow适应网格生成 adaptive grid generation 收缩 contraction守恒差分格式 conservation differences cheme 守恒型 conservation form数值边界条件 numerical boundary condition 数值耗散 numerical dissipation数值模拟 numerical simulation数值色散 numerical dispersion数值通量 numerical flux数值网格生成 numerical grid generation数值粘性 numerical viscosity数植扩散 numerical diffusion水动[力]噪声 hydrodynamic noise水动力学 hydrodynamics水洞 water tunnel水击 water hammer水静力学 hydrostatics水力半径 hydraulic radius水力坡度 hvdraulic slope水力学 hydraulics水头损失 head loss水位 water level水翼 hydrofoil水跃 hydraulic jump丝线法 tuft method斯坦顿管 Stanton tube斯坦顿数 Stanton number斯托克斯波 Stokes wave速度[水]头 velocity head速度环量 velocity circulation速度亏损律 velocity defect law速度剖面 velocity profile速度势 velocity potential随机选取法 random choice method随体导数 material derivativeT泰勒不稳定性 Taylor instability泰勒数 Taylor number泰勒涡 Taylor vortex特征线法 method of characteristics贴体曲线坐标 body- fitted curvilinear coordi-nates 通量矢量分解法 flux vector splitting method通量校正传输法 flux-corrected transport method 头波 bow wave投影法 projection method湍流边界层 turbulent boundary layer湍流分离 turbulent separation拖曳水池 towing tank脱体激波 detached shock wave椭圆余弦波 cnoidal waveW蛙跳格式 leap-frog scheme外流 external flow完全气体 perfect gas网格雷诺数 cell Reynolds number微压计 micromanometer维数分解 dimensional split尾流 wake [flow]未扰动流 undisturbed flow位移厚度 displacement thickness温度边界层 thermal boundary layer文丘里管 Venturi tube纹影法 schlieren method涡 eddy涡层 vortex layer涡对 vortex pair涡方法 vortex method涡管 vortex tube涡环 vortex ring涡街 vortex street涡量 vorticity涡量方程 vorticity equation涡量计 vorticity meter涡量拟能 enstrophy涡面 vortex surface涡片 vortex sheet涡丝 vortex filament涡线 vortex line涡旋 vortex涡旋破碎 vortex breakdown涡旋脱落 vortex shedding涡粘性 eddy viscosity无反射边界条件 nonreflecting boundary condition 无滑移条件 non-slip condition无量纲参数 dimensionless parameter无旋流 irrotational flow无压流 free surface flow无粘性流体 nonviscous fluid, inviscid fluid物理解 physical solution物理区域 physical domainX吸出 suction希尔特稳定性分析 Hirt stability analysis稀疏波 rarefaction wave稀疏矩阵分解法 split coefficient matrix method 细长度 slenderness细长体 slender body显格式 explicit scheme相容条件 consistency condition相似理论 similarity theory相似律 similarity law相似性解 similar solution响应频率 response frequency消能 energy dissipation楔流 wedge flow斜激波 oblique shock wave斜迎风格式 skew-upstream scheme谢齐公式 Chezy formula形状因子 shape factor型阻 profile drag修正微分方程 modified differential equation 旋臂水池 rotating arm basinY压[力]降 pressure drop压[强水]头 pressure head压差 differential pressure压差阻力 pressure drag压力能 pressure energy压强传感器 pressure transducer压强计 manometer压缩波 compression wave亚声速流[动] subsonic flow烟丝法 smoke wire method衍射 diffraction堰流 weir flow叶栅流 cascade flow液体动力润滑 hydrodynamic lubrication液体动力学 hydrodynamics液体静力学 hydrostatics液体静压 hydrostatic pressure依赖域 domain of dependence异重流 density current, gravity flow翼弦 chord翼型 airfoil阴影法 shadow method隐格式 implicit scheme迎风格式 upstream scheme , upwind scheme 迎角 angle of attack影响域 domain of influence壅水曲线 back water curve涌波 surge wave油膜显示 oil film visualization油烟显示 oil smoke visualization有限体积法 finite volume method有旋流 rotational flow有压流 pressure flow诱导速度 induced velocity诱导阻力 induced drag预估校正法 predictor-corrector method源 source远场边界条件 far field boundary condition 远场流 far field flow约束涡 confined vortex匀熵流 homoentropic flow运动相似 kinematic similarity运动粘性 kinematic viscosityZ再层流化 relaminarization再附 reattachment暂态流 transient flow折射 refraction振荡流 oscillatory flow蒸发 evaporation正激波 normal shock wave支配方程 governing equation直线法 method of lines指数格式 exponential scheme质点法 particle method质点网格法 particle in cell method,PIC method 质量传递 mass transfer质量守恒 conservation of mass滞后 lag滞止流 stagnation flow重力波 gravity wave周期流 periodic flow轴对称流 axisymmetric flow注入 injection驻点 stagnation point驻涡 standing vortex状态方程 equation of state锥形流 conical flow准定常流 quasi-steady flow准谱法 pseudo-spectral method自动网格生成 automatic grid generation [自]适应网格 [self-] adaptive mesh自由对流 natural convection, free convec-tion 自由流 free stream自由流线 free stream line自由面 free surface自由射流 free jet总焓 total enthalpy总压[力] total pressure总压头 total head阻力 drag, resistance阻尼误差 damping error。

剪应力计算方法

剪应力计算方法

Mathematical FormulationThe wall -function approach in ANSYS CFX is an extension of the method of Launder andSpalding [13]. In the log-law region, the near wall tangential velocity is related to the wall -shear -stress, , by means of a logarithmic relation.In the wall -function approach, the viscosity affected sublayer region is bridged by employing empirical formulas to provide near-wall boundary conditions for the mean flow and turbulence transportequations. These formulas connect the wall conditions (e.g., the wall -shear -stress) to the dependent variables at the near-wall mesh node which is presumed to lie in the fully-turbulent region of theboundary layer.The logarithmic relation for the near wall velocity is given by: where:u + is the near wallvelocity, is the friction velocity, U t is the known velocity tangent to the wallat a distance of from the wall , y+ is the dimensionless distance from the wall ,is thewall shear stress, is the von Karman constant and C is a log-layer constant depending on wallroughness (natural logarithms are used).A definition ofin the different wallformulations is available. For details, see Solver Yplus andYplus. Scalable Wall Functions Equation 164 has the problem that it becomes singular at separation points where the near wall velocity, U t , approaches zero. In the logarithmic region, an alternative velocity scale, u* can be used instead of u +:This scale has the useful property that it does not go to zero if U t goes to zero (in turbulent flow k is never completely zero). Based on this definition, the following explicit equation forcan beobtained: The absolute value of the wall shear stress , is then obtained from:where: and u * is as defined earlier.Modeling Flow Near the WallPrev Up / Home Wall Distance Formulation NextEquation 164.Equation 165.Equation 166.Equation 167.Equation 168.Equation 169.Equation 170.ANSYS CFX-Solver Theory Guide | Turbulence and Wall Function Theory | Modeling Flow Near the Wall |One of the major drawbacks of the wall -function approach is that the predictions depend on the location of the point nearest to the wall and are sensitive to the near-wall meshing; refining the mesh does not necessarily give a unique solution of increasing accuracy (Grotjans and Menter [10]). The problem of inconsistencies in the wall -function, in the case of fine meshes, can be overcome with the use of the Scalable Wall Function formulation developed by ANSYS CFX. It can be applied on arbitrarily fine meshes and allows you to perform a consistent mesh refinement independent of the Reynolds number of the application.The basic idea behind the scalable wall -function approach is to limit the y * value used in the logarithmic formulation by a lower value of . 11.06 is the intersection between thelogarithmic and the linear near wall profile. The computed is therefore not allowed to fall below this limit. Therefore, all mesh points are outside the viscous sublayer and all fine mesh inconsistencies are avoided.It is important to note the following points:●To fully resolve the boundary layer, you should put at least 10 nodes into the boundary layer.●Do not use Standard Wall Functions unless required for backwards compatibility.●The upper limit for y + is a function of the device Reynolds number. For example, a large ship may have a Reynolds number of 109 and y + can safely go to values much greater than 1000. For lowerReynolds numbers (e.g., a small pump), the entire boundary layer might only extend to aroundy + = 300. In this case, a fine near wall spacing is required to ensure a sufficient number of nodes in the boundary layer.If the results deviate greatly from these ranges, the mesh at the designated Wall boundaries will require modification, unless wall shear stress and heat transfer are not important in the simulation.Solver Yplus and YplusIn the solver output, there are two arrays for the near wall spacing. The definition for the Yplusvariable that appears in the post processor is given by the standard definition ofgenerally used in CFD:where is the distance between the first and second grid points off the wall .In addition, a second variable, Solver Yplus, is available which contains the used in the logarithmic profile by the solver. It depends on the type of wall treatment used, which can be one of three different treatments in ANSYS CFX. They are based on different distance definitions and velocity scales. This has partly historic reasons, but is mainly motivated by the desire to achieve an optimum performance in terms of accuracy and robustness:●Standard wall function (based on ) ●Scalable wall function (based on ) ●Automatic wall treatment (based on )The scalable wall function y + is defined as:and is therefore based on 1/4 of the near wall grid spacing.Note that both the scalable wall function and the automatic wall treatment can be run on arbitrarily fine meshes.Equation 171.Equation 172.Automatic Near-Wall Treatment for Omega-Based ModelsWhile the wall -functions presented above allow for a consistent mesh refinement, they are based on physical assumptions which are problematic, especially in flows at lower Reynolds numbers (Re<105), as the sublayer portion of the boundary layer is neglected in the mass and momentum balance. For flows at low Reynolds numbers, this can cause an error in the displacement thickness of up to 25%. It is therefore desirable to offer a formulation which will automatically switch from wall -functions to a low-Re near wall formulation as the mesh is refined. Themodel of Wilcox has the advantage that an analytical expression is known for in the viscous sublayer, which can be exploited to achieve this goal. The main idea behind the present formulation is to blend the wall value for between thelogarithmic and the near wall formulation. The flux for the k -equation is artificially kept to be zero and the flux in the momentum equation is computed from the velocity profile. The equations are as follows: Flux for the momentum equation, F U :with:Flux for thek -equation: In the -equation, an algebraic expression is specified instead of an added flux. It is a blend betweenthe analytical expression forin the logarithmic region: and the corresponding expression in the sublayer:with being the distance between the first and the second mesh point. In order to achieve a smoothblending and to avoid cyclic convergence behavior, the following formulation is selected: While in the wall -function formulation, the first point is treated as being outside the edge of the viscous sublayer, the location of the first mesh point is now virtually moved down through the viscous sublayer as the mesh is refined in the low-Re mode. It is to be emphasized, that the physical location of the first mesh point is always at the wall (y = 0). The error in the wall -function formulation results from this virtual shift, which amounts to a reduction in displacement thickness. This error is always present in the wall -function mode, but is reduced to zero as the method shifts to the low-Re model. The shift is based on the distance between the first and the second mesh point y = y 2 - y 1 with y being the wall normal distance. Treatment of Rough WallsThe above wall function equations are appropriate when the walls can be considered as hydraulically smooth. For rough walls , the logarithmic profile still exists, but moves closer to the wall . RoughnessEquation 173.Equation 174.Equation 175.Equation 176.Equation 177.Equation 178.Equation 179.effects are accounted for by modifying the expression for u + as follows:where:and y R is the equivalent sand grain roughness [14].You are advised to exercise care in the use of the rough wall option together with the standard wall -function approach. Inaccuracies can arise if an equivalent sand grain roughness is of the same order, orlarger than the distance from the wall to the first node.The first element off thewall should not be much thinner than the equivalent sand grain roughness. If this element becomes too thin, then negative values of u +can be calculated, resulting in the ANSYS CFX-Solver failing. For details, see Wall Roughness.Heat Flux in the Near-Wall RegionThe thermal boundary layer is modeled using the thermal law-of-the-wallfunction of B.A. Kader [15]. Heat flux at the wall can be modeled using a wall function approach or the automatic wall treatment. Using similar assumptions as those above, the non-dimensional near-wall temperature profile follows a universal profile through the viscous sublayer and the logarithmic region. The non-dimensionaltemperature, T +, is defined as:where T w is the temperature at the wall , T f the near-wall fluid temperature, c p the fluid heat capacity and q w the heat flux at the wall . The non-dimensional temperature distribution is then modeled as:where: Pr is the fluid Prandtl number, given by:where is the fluid thermal conductivity. Combining these equations leads to a simple form for the wall heat flux model:Turbulent fluid flow and heat transfer problems without conjugate heat transfer objects require the specification of the wall heat flux, q w , or the wall temperature, T w . The energy balance for eachboundary control volume is completed by multiplying the wall heat flux by the surface area and adding to the corresponding boundary control volume energy equation. If the wall temperature is specified, the wall heat flux is computed from the equation above, multiplied by the surface area and added to the boundary energy control volume equation.Equation 180.Equation 181.Equation 182.Equation 183.Equation 184.Equation 185.Equation 186.Equation 187.Additional VariablesThe treatment of additional scalar variables in the near wall region is similar to that for heat flux.Treatment of Compressibility EffectsWith increasing Mach number (Ma > 3), the accuracy of the wall -functions approach degrades, which can result in substantial errors in predicted shear stress, wall heat transfer and wall temperature for supersonic flows.It has been found that the incompressible law-of-the-wall is also applicable to compressible flows if the velocity profile is transformed using a so-called ”Van Driest transformation” [16]. The logarithmic velocity profile is given by:where , , = 0.41 and C = 5.2. The subscript w refers to wall conditions, and the subscript “comp” refers to a velocity defined by the following equation(transformation):Near a solid wall , the integrated near-wall momentum equation reduces to:while the energy equation reduces to: Expressions for shear stress and heat flux applicable to the boundary layer region are:and If Equation 190, Equation 192 and Equation 193 are substituted into Equation 191 and integrated, the resulting equation is:which provides a relationship between the temperature and velocity profiles. Using the perfect gas law, and the fact that the pressure is constant across a boundary layer, Equation 194 replaces the density ratio found in Equation 189. Performing the integration yields the following equation for the “compressible” velocity:where:Equation 188.Equation 189.Equation 190.Equation 191.Equation 192.Equation 193.Equation 194.Equation 195.Equation 196.Equation 197.Equation 198.The above derivation provides most of the equations necessary for the implementation ofwall -functions which are applicable to compressible flows. These wall-functions are primarily implemented in two locations in ANSYS CFX: hydrodynamics and energy. First, consider the implementation in thehydrodynamics section. The equation for the wall -shear -stress is obtained after a slight rearrangement of Equation 188:This is similar to the low speed wall -function, except that U compnow replaces U . The Van Driesttransformation given byEquation 195, must now be performed on the near wall velocity. In the implementation, Equation 195 is re-written in the following form:where: This completes the wall -function modifications for the hydrodynamics.The relationship between wall heat transfer, near wall velocity and near wall temperature is given by Equation 194 (and also in rearranged form after Equation 200: the equation for T w / T ). A dimensionless variable, , can be defined as:and hence: From Equation 203 it is clear that knowing , the near wall velocity, the near wall temperature and the wall heat transfer, the wall temperature can be calculated (if instead the wall temperature is known, then the wall heat transfer is calculated). Before proceeding, it is instructive to consider the equationEquation 199.Equation 200.Equation 201.Equation 202.Equation 203.that is used in ANSYS CFX for low Mach number heat transfer. The equation is subsequently modified for use at higher Mach numbers. This unmodified equation is given by:where: It can be seen that this equation blends between the linear and logarithmic near wall regions, and hence is more general than just using the logarithmic profile that is implied byEquation 202. Equation 204 has been extended for use in the compressible flow regime by interpreting the linear and logarithmicvelocity profiles given above as “compressible” or Van Driest transformed velocities:This change is consistent withEquation 188. The “untransformed” velocities required by Equation 204can be obtained by applying the inverse Van Driest velocity transformation to these “compressible” velocities. The inverse transformation is given by:where:and A and B are defined followingEquation 195 above. The reverse transformed velocities obtained from Equation 211 are substituted intoEquation 204 to obtain the value of. Either the wall heat transfer or the wall temperature can then be obtained from Equation 203.Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates.Equation 204.Equation 205. Equation 206.Equation 207.Equation 208.Equation 209. Equation 210.Equation 211.Equation 212.Equation 213.PrevModeling Flow Near the Wall Up / Home Next Wall Distance Formulation。

壁面函数

壁面函数

FLUENT壁面函数的选择2011-10-09 10:22:05| 分类：默认分类|举报|字号订阅壁面函数问题1、无论是标准k—ε模型、RNGk—ε模型，还是Realizable k—ε模型,都是针对充分发展的湍流才有效的，也就是说，这些模型均是高Re数的湍流模型。

它们只能用于求解处于湍流核心区的流动。

而壁面函数是对近壁区的半经验描述，是对某些湍流模型的补充（近壁区对整体流动影响较大和低雷诺数Re的情况），通过壁面函数法和低Re数k—ε模型与标准k—ε模型和RNGk—ε模型配合，成功解决整个整个管道的流动计算问题。

2、在壁面区，流动情况变化很大。

解决这个问题目前有两个途径：一、是不对粘性影响比较明显的区域（粘性底层和过渡层）进行求解，而是用一组半经验的公式（即壁面函数）将壁面上的物理量与湍流核心区内的相应物理量联系起来。

这就是壁面函数法。

在划分网格的时候，不需要在壁面区加密，只需要把第一个节点布置在对数律成立的区域内，即配置在湍流充分发展区域。

如果要用到壁面函数的话，在define---modle--viscous面板里有near wall treatment一项。

可以选择标准壁面函数、不平衡壁面函数等。

二、是采用低Re数的k—ε模型来求解粘性底层和过渡层，此时需要在壁面区划分比较细密的网格，越靠近壁面，网格越细。

当局部湍流的Re数小于150时，就应该使用低Re数的k—ε模型。

总结：相对于低Re数的k—ε模型，壁面函数法计算效率高，工程实用性强。

但当流动分离过大或近壁面流动处于高压之下时，不是很理想。

在划分网格的时候，需要在壁面的位置设置边界层网格，原因也是如此。

====================================================== ============================================为什么要用壁面函数？？就是因为，k-epsilon模型中，k的boundary condition已知，在壁面上为零，而epsilon的boundary condition 在壁面上为一未知的非零量，如此如何来解两方程模型所以，我们就需要壁面函数来确定至少第一内节点上的值，当然也包括壁面上的值。