A GAS-KINETIC SCHEME FOR TURBULENT FLOW

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1 A GAS-KINETIC SCHEME FOR TURBULENT FLOW M. Righi Zurich University of Applied Sciences Abstract A gas-kinetic scheme is used in this study to simulate turbulent flow and the interaction between a shock wave and a turbulent boundary layer in particular. Gas-kinetic schemes belong to a class of their own of numerical schemes for fluid mechanics. These schemes are neither particle methods, such as DSMC and Molecular Dynamics, nor Lattice Boltzmann methods: they calculate the fluxes between numerical cells on the basis of the Boltzmann equation instead of the Navier-Stokes or Euler equations. Advective and viscous fluxes are coupled in gas-kinetic schemes; in other words, the effect of unresolved fluctuations (represented in conventional schemes by the viscous fluxes) may affect the transport terms (advection). In simulations of turbulent flow where the unresolved turbulent fluctuations are modeled by eddy viscosity, a gas-kinetic scheme may exploit the information provided by the turbulence model in a different way. The analysis proposed in this paper shows that gas-kinetic schemes have an in-built multiscalar sensor which controls the correction terms generated by the underlying kinetic theory. The simulations shown in this paper have been obtained with the RANS approach and a standard two-equation turbulence model (k-ω). It is shown that the gas-kinetic scheme provides good quality predictions in a number of test cases, where conventional schemes with the same turbulence model are known to fail. Introduction Turbulence is clearly a multiscale problem. In numerical simulations, spatial and temporal resolution may be insufficient to resolve all turbulent scales of motion down to the dissipative range. The effects of the unresolved scales of motion on the resolved ones must then be modeled. This approach has been inspired by the handling of thermal fluctuations, which can virtually never be resolved in the continuum regime, which are invariably modeled by diffusion proportional to a given viscosity. However, unlike thermal fluctuations, unresolved turbulent fluctuations may not be small and spread over a range without a clear separation from the resolved scales of motion. In theory, this situation, which is even a pre-requisite in Large Eddy Simulation, may make conventional advection-diffusion schemes physically inconsistent. A similar boundary is reached when (laminar) rarefied flows are simulated with conventional Navier-Stokes schemes. In rarefied flows, it may happen that even the scale of thermal fluctuations become not negligible with respect to the resolved scales. The unresolved-to-resolved scales ratio assumes large values in special flow regions, such as shocklayers, due to presence of strong gradients which may lead to small resolved scales and a high turbulence intensity, which may be related to large (in space and time) turbulent scales of motion. It is well accepted that conventional schemes, with the RANS approach in particular, may fail to accurately capture details of the shock-boundary layer interaction such as strength and position of the shock, position of separation and re-attachment, turbulence intensity downstream of the separated flow region. A comprehensive review can be found in [Babinsky and Harvey, 0]. In many cases, such as the design of supersonic or hypersonic vehicles, inaccurate prediction of this area may lead to unsuitable geometries or high development costs. In this study a gas-kinetic scheme is used, following the RANS approach, to simulate turbulent flow. Gas-kinetic schemes are derived form the Boltzmann equation and not from the Navier-Stokes or Euler equations. They are suitable for finite volume or finite elements schemes but, being relatively little known, they are often confused with particle methods such as DSMC or Molecular Dynamics or with Lattice Boltzmann methods. Numerous gas-kinetic schemes have been developed over the latest twenty years [Xu, 00, May et al., 007, Mandal and Deshpande, 99, Chou and Baganoff, 997, Xu and Prendergast, 99] with the aim to achieve a physically more consistent mathematical model of fluid mechanics. Gas-kinetic schemes are more accurate than conventional schemes, and might be able to resolve shock-layers. Besides, they are more suitable to high-order reconstruction [Li et al., 00, Xu et al., 005, Xuan and Xu, 0] and may be used as a platform to investigate rarefied flow [Liao et al., 007, Xu and Huang, 00]. Since conservation laws such as the Navier-Stokes and Euler equations can be derived from the Boltzmann equation, as shown in [Cercignani, 988, Xu, 998], a gas-kinetic scheme is always consistent with a conventional one, within the validity boundaries of the latter. Simulation of rarefied flow implies some multiscalar mechanism, i.e. the ability to correct the

2 conventional advective and diffusive fluxes whenever the unresolved scales of motion are not negligibly small. This property might prove interesting in the simulation of turbulent flow. To build the turbulent gas-kinetic scheme shown in this paper, the scheme developed by [Xu, 00] has been the starting point. An allied two-equation turbulence model (k-ω) is solved alongside the equations for the conservative variables (with Boltzmann-BGK fluxes). The strength of the scheme lies in its capability to evaluate advective and diffusive fluxes in a single operation, including therefore the effect of collisions on transport, or, in other words, to take into account the scale of the unresolved fluctuations - introducing a multiscale effect which is missing in Navier-Stokes schemes. Note that the inclusion of kinetic effects also generates a misalignment between turbulent stress and strain rate acting in a similar way to high-order turbulence models, as remarked in [Chen et al., 00]. The structure of this draft paper includes the description of the GKS used and modified for turbulent flows in section, the presentation of results obtained in numerical experiments and conclusions. Gas-kinetic schemes The state of a gas can be described by means of a distribution function f(x, v, t) - defined in the phase space. The conservative variables w = [ρ ρv ρv ρv ρe] T are recovered from f by taking moments in the phase space: w = ψfdξ, () where the elementary volume in phase space is dξ = dv dv dv dξ and: ψ = [ ( v v v ui + ξ )] T. () The numerical fluxes F related to a unit interface length normal to direction n, and a time step t are obtained by time integration: F n = t 0 fψu n dξ dt. () The distribution function f is assumed to be a solution of the Boltzmann-BGK equation [Bhatnagar et al., 95]: f t + (u )f = f eq f, () τ where τ is a relaxation time corresponding to the average fluctuation period and taken to be proportional to molecular viscosity and inversely proportional to static pressure in laminar flow, τ = µ p. A distribution function consistent with Eq. is obtained by means of the Chapman-Enskog expansion [Cercignani, 988]: f = f eq ɛ τdf eq + ɛ τd ( τdf eq ) (5) where τ is a reference relaxation time and ɛ is a small number representing the deviations from thermal equilibrium. It can be demonstrated [Cercignani, 988, Xu, 998] that Eq. 5 truncated at the first order ignores thermal fluctuations and is consistent with Euler fluxes. Eq. 5 truncated at the second order would generate Navier-Stokes fluxes whereas the next two truncation orders would provide the Burnett and super Burnett equations. The scheme described in this paper is a modification of the one developed in [Xu, 00] for laminar flow. It uses the MUSCL approach and assumes as such a conventional discontinuous reconstruction. The gas evolution is integrated in time at each time step. An initial distribution f 0 is generated on the basis of the gas states and adopting a second-order Chapman-Enskog expansion. Further, a distribution function f BGK is obtained as the analytical solution to the BGK equation, Eq. : τ f BGK (x, x, x, t, u, u, u, ξ) = t o f eq (x, x, x, t, u, u, u, ξ)e (t t )/τ dt +e t/τ f 0 (x u t, x u t, x u t) (6), where x = x u (t t ), x = x u (t t ), x = x u (t t ). The initial distribution f 0 is rather complex as it includes the spatial and temporal terms, necessary to obtain a proper Chapman-Enskog expansion. Details of the derivation can be obtained in [Xu, 00, Righi, 0, Righi, 0]. In this paper, the resulting distribution function, solution to the BGKmodel and compatible with the gas initial states, f BGK, is expressed in a compact form f BGK = f NS + e t/τ f, (7) where f NS is a second-order Chapman-Enskog expansion built around an average gas state at the interface, whereas f is a more complex function, obtained as difference between f NS and a Chapman- Enskog expansion obtained combining the left and right gas states. The presence of the blending function e t/τ is important and represents the kernel of the multiscalarity of the gas-kinetic scheme. It is important to remark that f NS generates fluxes which might be compared to the ones provided by a central conventional scheme. At first sight, Eq. 7, has the typical structure of upwind schemes: central fluxes plus a correction term whose role becomes more important the stronger the gradients involved. The fluxes are finally obtained inserting f = f BGK into Eq. ; they can be expressed into the form: F n = F NS n + α(ɛ) F n, (8)

3 where Fn NS are the fluxes associated with f NS and Fn the ones associated with f, α(ɛ) = ɛ( e /ɛ )) and ɛ = τ/ t. In Eq. 8 the parameter ɛ plays a very important role, as it controls the correction kinetic terms. Since ɛ can be seen as the ratio of unresolved to resolved timescales, the mechanism which adjusts the scheme to the size of unresolved scales is clearly visible. It is sometimes referred to as intrinsic multiscalarity of this scheme. This property has been exploited in laminar flow; since this scheme has also been used in the transitional regime [Liao et al., 007] with a suitable definition of τ. As is well known, the BGK model is consistent with conservationlaws with an unity Prandtl number; the heath flux must therefore be corrected for realistic fluids. A number of publications ([Li et al., 00, Xu et al., 005, Xuan and Xu, 0] and references therein) demonstrate the good qualities of the scheme. Following the RANS approach, the effect of unresolved turbulence can be modeled through an eddy viscosity µ t, determined by the modeled turbulent quantities. In a gas-kinetic scheme, the relaxation time τ must be related to the eddy viscosity. Trivially, one could set τ = µ t /p in analogy with laminar flow. However, the considerations exposed in [Chen et al., 00, Chen et al., 00] lead to: k/ω τ = µ p +, (9) T ( + η / ) where T is temperature and η = Sk/ε, S is a measure of the local velocity gradient. The turbulent quantities k and ω are provided by a standard k-ω model solved alongside. In turbulent flow, the quantity ɛ = τ/ t may assume large values, leading to numerical problems. It is proposed to replace it with: ɛ =, (0) τ τ where τ is the assumed timescale of the resolved flow and can be calculated form the gradients of one of the resolved variables (i.e. τ = ρ/dρ). Details are explained in [Righi, 0]. The expression of ɛ used in Eq. 0 is close to the one used in rarefied gas dynamics to estimate the local Knudsen number, with the difference that in this case timescales are used instead of spatial quantities. In the section dedicated to the results obtained, it can be seen that this timescale ratio or degree of rarefaction may assume values up to a few hundredths inside shocklayers. In rarefied gas dynamics the validity of Navier-Stokes schemes may become questionable for Knudsen number as high as one thousandth. The use of a gas-kinetic scheme with eddy viscosity may therefore exploit its better ability to cope with these rarefaction levels. Numerical implementation The gas-kinetic scheme described above has been implemented into a two-dimensional finite-volume steady-state solver. The results shown in this paper have been obtained with a second-order reconstruction. The minmod limiter has been used for the reconstruction of conservative variables and their gradients. The gas-kinetic scheme has been used in this study with CFL between and 0, which is slightly higher than the value used with the same solver with conventional fluxes. This not withstanding, the overall computational cost requested by the gas-kinetic scheme is higher by a factor of two on average. The solver is sequential and no attempts have been done to optimize the code for speed. No wall functions have been used, the viscous sublayer has been accurately resolved in every flow case. Results and Discussion In this paper results from four test cases are shown: the flow around the RAE8 supercritical airfoil at the conditions of Case 0 in the experiments published in [Cook et al., 979], the supersonic compression ramp investigated in [Settles et al., 979] and the one at lower Reynolds number investigate in [Bookey et al., 005], the hypersonic compression ramp investigated in [Dolling and Erengil, 99]. Transonic flow around a RAE 8 airfoil in Case 0 In Case 0 the airfoil is immersed in a flow at Mach M = 0.75 at an angle of attack α =.9 and Reynolds number Re c = based on airfoil chord. The experimental investigation have highlighted an incipient separation of the boundary layer, which fails to be predicted by most conventional solvers with a linear twoequation turbulence model (refer to the discussion in [Wallin and Johansson, 000]). More sophisticated turbulence models such as algebraic stress models may capture the incipient separation correctly but the calibration of all coefficients is seldom suitable for many flow classes. Fig. shows the results obtained with the gas-kinetic scheme and the ones obtained with a conventional solver based on MUSCL reconstruction and Roe s approximate Riemann solver. Remarkably, the gas-kinetic scheme provides a rather accurate prediction, whereas the conventional Navier-Stokes scheme places the shock slightly downstream of the position indicted by the measurements. Supersonic compression corner at high and low Reynolds number The supersonic flow impinging on a compression corner has been studied by many investigators in the past. One of the most popular campaign is the one carried out at Princeton and published in [Settles et al., 976, Settles et al., 979]. The flow is characterized by a Mach M =.85 and a Reynolds number Re θ = 000, based on momentum thick-

4 Cp Cf (a) x/c (b) x/c Figure : RAE8 airfoil (Case 0 Re = , M = 0.75, angle of attack α =.9 ). ( ) Gas-kinetic scheme (GKS) on finest grid, ( ) GKS on medium grid, ( ) GKS on coarsest grid, ( ) Navier-Stokes (Roe s approximate Riemann solver) on finest grid, ( o ): experimental data from Cook [Cook et al., 979]. (a) pressure coefficient, (b) skin friction coefficient. ness. Four different corners have been used: 8, 6, 0 and. The 8 corner does not separate the flow and the 6 one generates only an incipient separation. The time-averaged separation at 0 spans about δ and the one at about δ (where δ is the incoming boundary layer thickness). Results from conventional Navier-Stokes scheme are not reported in this paper, as for these flow cases, they tend to become more dependent on numerical details such as reconstruction technique and factorization. In order to avoid proposing one particular implementation, the reader is referred to the literature, for instance on the comprehensive review in [Babinsky and Harvey, 0], [Goldberg et al., 998] and [Menter and Rumsey, 99]. As is well accepted, conventional schemes fail to predict the right shock position and separation length: the whole interaction region is often translated downstream, the extension of the separation is often underestimated. The turbulent gas-kinetic scheme developed in this study does not provide a perfect agreement with experiments but does better than conventional schemes. Figures and show the reasonably good agreement of predictions in all four cases in terms of wall pressure and skin friction coefficient. In order to generate data for a benchmark case for LES and DNS methods, the same flow, at much a lower Reynolds number of Re θ = 00, has been investigated many years later also in Princeton and published in [Bookey et al., 005]. The first experiment of this campaign concerns a supersonic compression corner with an angle of. Within the same investigation an shock impinging with an angle of onto a turbulent boundary layer has also been investigated. The two flow generates a similar shock-boundary layer interaction. Fig. shows the pressure distributions for both the compression corner and the impinging shock, which are in reasonably good agreement with the experiments. For both flow cases, the one investigated by Settles and the one investigated by Bookey, the sensitivity to the Reynolds number has been investigated with the gas-kinetic scheme. Fig. 5 shows the interaction length as well as the separation point as a function of the Reynolds number. Both quantities are compared to the values estimated with an empirical law devised by Settles, details can be found in [Délery et al., 986]. The prediction is very accurate at lower Reynolds number and barely acceptable in the other case. Moreover, the effect of wall temperature on the interaction has also been investigated with the gaskinetic scheme. In this case the experimental data published in [Spaid and Frishett, 97] has been used. Fig. 6 shows the separation length and the separation

5 α = 8.0 Cf 0 α = 8 α = 6.0 Cf 0 α = 6 α = 0.0 Cf 0 α = 0 α =.0 Cf 0 α = Figure : Pressure calculated for four different compression corner flows, characterized by angles values of 8, 6, 0 and (freestream conditions: M =.85, Re = per length unit, δ 0 = m). ( ) Gas-kinetic scheme (GKS) on finest grid, ( ) GKS on medium grid (only shown for α = ), ( ) GKS on coarsest grid (only shown for α = ), ( o ): experimental data from Settles [Settles et al., 979]. Figure : Skin friction calculated for four different compression corner flows, characterized by angles values of 8, 6, 0 and (freestream conditions: M =.85, Re = per length unit, δ 0 = m). ( ) Gas-kinetic scheme (GKS) on finest grid, ( ) GKS on medium grid (only shown for α = ), ( ) GKS on coarsest grid (only shown for α = ), ( o ): experimental data from Settles [Settles et al., 979].

6 5 (a) L0/δ e + 06 e + 06 e + 06 Re.00 (b).50 L0/δ e + 07 e + 08 e + 08 Re Figure : Distribution of static pressure calculated for the compression corner (a) with an angle of and for the reflected shock (b) originating form a compression corner at. Freestream conditions (both flows) are M =.90, Re θ = 00. ( ) GKS on grid, ( ) GKS on grid, ( o ): experimental data from Bookey [Bookey et al., 005]. Grid sizes compression corner: 8 9, 5 68 respectively, grid size impinging shock: 8 08, 96 0 respectively. Grid and have different resolution and have been generated with different algorithms. point as a function of the wall temperature. The prediction can be considered as acceptable. The distribution of the time scales ratio ɛ is shown in the Figures 7 and 8. In both cases, it is evident that ɛ assumes the larger values in the interaction area, that is, where the results obtained with the gas-kinetic scheme and conventional schemes differ most. Hypersonic compression corner at Mach.95 This flow case, investigated by Dolling et al. [Dolling and Erengil, 99], extends the Mach number range to the hypersonics. However, adiabatic wall conditions are assumed. Reynolds number is based on boundary layer thickness. Results from RANS and hybrid simulations can be found in Edwards et al. [Edwards, 008]. In Fig. 9 the pressure distribution predicted by the turbulent gaskinetic scheme is compared to the experimental values [Dolling and Erengil, 99], showing an acceptable agreement. In Fig. 9 the distribution of the time scales ratio ɛ is shown. It is evident that the highest Figure 5: Interaction length as a function of Reynolds number; the prediction obtained with the gas-kinetic scheme are compared to the empirical law by Settles from [Délery et al., 986]. values arise in the interaction area, that is where the results provided by the gas-kinetic scheme and conventional schemes differ most. 5 Conclusions In this paper a gas-kinetic scheme is used to simulate turbulent flow, following the RANS approach with a standard two-equation model. The test cases proposed contain strong interactions between a shock and the turbulent boundary layer, causing a substantial separation. It is shown that the results obtained with gaskinetic scheme - unlike the ones, available in the literature, obtained with conventional schemes - are in good or very good agreement with the experimental data. The rationale is put forward, that the higher accuracy of the scheme proposed originates form the multiscalar mechanisms, a legacy of the kinetic theory of gases, which corrects the numerical fluxes as a function of the unresolved-to-resolved scales ratio. The idea of eddy viscosity assumes that the unresolved scales of motion (turbulent fluctuations) are small and well separated from the smallest resolved scales. As is well known, this condition is not always met in and around shocklayers, where the resolved scales are small and turbulence level is high. In theory, this situation might make conventional schemes unsuitable; on the other hand, it might provide a clear advantage for gas-kinetic schemes.

7 xm/δ T w /T t xs/δ T w /T t Figure 6: Effects of wall temperature for a 0 angle ramp, experimental data from [Spaid and Frishett, 97]. T w is wall temperature, T t is stagnation temperature, x m is the starting point of interaction from the corner, x s is the starting point of separation from the corner. Figure 9: Compression corner M =.95, Experimental values from Dolling et al. [Dolling and Erengil, 99]. Results obtained on a grid with size: 8 60 The behavior of gas-kinetic schemes in turbulent simulations needs additional validation, in particular in three dimensional geometries. However, the results obtained so far are very encouraging; two-dimensional test cases are proposed ranging from transonic to hypersonic (M = 5) where the gas-kinetic scheme predicts position and size of the interaction in an acceptable way. It is worthwhile observing that the same approach might be suitable for Large Eddy Simulation too, as the cut-off scale in LES is by definition in the inertial subrange, i.e. between adjacent smallest-resolved and largest-unresolved scales. 6 References Figure 7: Compression corner at higher Reynolds number investigated in [Settles et al., 976], distribution of the time scales ratio ɛ. [Babinsky and Harvey, 0] Babinsky, H. and Harvey, J. K. (0). Shock wave-boundary-layer interactions. Cambridge University Press New York. [Bhatnagar et al., 95] Bhatnagar, P., Gross, E., and Krook, M. (95). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev., 9():5 55. Figure 8: Compression corner at lower Reynolds number investigated in [Bookey et al., 005], distribution of the time scales ratio ɛ. [Bookey et al., 005] Bookey, P., Wyckham, C., and Smits, A. (005). Experimental investigations of Mach shock-wave turbulent boundary layer interactions. AIAA Paper No [Cercignani, 988] Cercignani, C. (988). The Boltzmann equation and its applications. Springer, New York.

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