An assessment of twoequation turbulence models, the low Reynolds kε and kω SST models, with the compressibility corrections proposed by Sarkar and Wilcox, has been performed. The compressibility models are evaluated by investigating transonic or supersonic flows, including the arcbump, transonic diffuser, supersonic jet impingement, and nsteady supersonic diffuser. A unified implicit finite volume scheme, consisting of mass, momentum, and energy conservation equations, is used, and the results are compared with experimental data. The model accuracy is found to depend strongly on the flow separation behavior. An MPI (Message Passing Interface) parallel computing scheme is implemented.
1. Introduction
Supersonic propulsion systems have applications for a wide variety of aircraft and rockets, operating at flow regimes from subsonic to supersonic. A variety of flow devices for the vehicles operate at a wide range of Mach numbers, so that shock / boundary layer interactions are common, and may have a significant influence on the entire flow field
[1

5]
.
At high Reynolds numbers, flows are very likely to be fully turbulent. With increasing turbulence Mach number, the velocity fields can no longer be assumed to be incompressible. For the transonic and supersonic flow regimes, the shock / boundary layer interaction causes a large increase in turbulence intensity and shear stress. Turbulence modeling for compressible flow has to account for the additionalcorrelations, involving both the fluctuating thermodynamic quantities, and the fluctuating dilatation. To account for the important flow features at transonic and supersonic conditions, the combined models of compressibledissipation and pressuredilatation proposed by Sarkar and Wilcox
[6

9]
were evaluated in this study.
In smooth flow regions with continuous flow variables, the centraldifferencing schemes based on the Taylor series expansion can be applied, with a certain order of accuracy. In the supersonic regime with discontinuities such as shock waves, however, the scheme becomes numerically unstable. To avoid such nonphysical oscillation, the concept of total variation diminishing (TVD) and the AUSMPW+ schemewere used for the inviscid flux calculation, as the upwind schemes
[10]
.
For this study, twoequation turbulence models, low Reynolds kε and kω SST models, with and without compressible dissipation and pressuredilatation corrections, are used to evaluate the compressibility effects of turbulent models in transonic and supersonic flows involving flow separation. The four test cases are a 7% arcbump
[11
,
12]
, transonic diffuser
[13
,
14]
, the supersonic impinging jet
[15]
, and the supersonic diffuser
[16
,
17]
.
2. Numerical Method
 2.1 Governing Equation
The Favreaveraged governing equations for a compressible flow, based on the conservation of mass, momentum, energy, and species, can be expressed as:
where, superscript “” represents time averaged quantity, and superscript “~” means mass weighted averaged quantity.
ρ, u, p, τ_{ij}, q_{j}, E
represent density, velocity, pressure, stress tensor, heat flux vector, and specific total internal energy, respectively.
t, x_{j}, δ
are time, space coordinate, and kronecker delta, respectively. Subscripts i and j represent space coordinates or tensor indices.
 2.2 Turbulent Models
The turbulent models considered for this study are twoequation turbulence models, such as the lowReynolds number kε model, and kω SST model.
The standard kε model was proposed for high Reynolds number turbulence flows, and is traditionally used with a wall function, and the variable y+ as a damping function. However, universal wall functions do not exist in complex flows, so the damping factor cannot be perfectly applied to the flow with separation. Thus, a low Reynolds number kε model was developed, to take account of nearwall turbulence. Shih and Lumley observed that, within certain distances from the wall, all energetic large eddies will reduce to Kolmogorov eddies; and all the important wall parameters, such as friction velocity, viscous length scale, and mean strain rate at the wall, can be characterized by the Kolmogorov micro scale
[18]
.
Yang and Shih proposed a timescalebased kε model for the nearwall turbulence, related to the Kolmogorov time scale as its lower bound, so that the equation can be integrated to the wall. The advantages of this model are the lack of a singularity at the wall, and adaptability to separated flow, since the damping function is based on the Reynolds number, instead of y+. The low Reynolds number models have been designed to maintain the high Reynolds number formulation in the loglaw region, and further tuned to fit measurements for the viscous and buffer layers
[19]
.
The turbulent kinetic energy and its dissipation rate are calculated from the turbulence transport equations, written as follows:
where
k, ε, μ, P_{k}, T_{t}
and ∧ are the kinetic energy, dissipation rate, viscosity, turbulent kinetic energy production rate, turbulent time scale, and damping function, respectively, and are then represented as follows:
where,
τ_{k}
is the Kolmogorov time scale.
The turbulent viscosity
μ_{t}
and damping factor
f_{μ}
for the wall effect can be written as
where,
R_{y}
is the damping factor, which is taken to be a function of
R_{y}=k^{1/2}y/v
.
The standard kω model predicts the boundary layer near the wall more accurately, but is sensitive to the initial condition of the free flow region. On the other hand, the standard kε model predicts the free flow region more accurately, but requires the wall function, and modeling near the wall. Therefore, Menter suggests a blended model, which uses the kω model at the inner region of a boundarylayer, and then uses the kε model for the outer boundary layer, and for free shear layers
[20]
.
For the kω SST mode, a blending function, F
_{1}
, is defined, so as to vary smoothly from a value of 1 near a wall, through the loglaw region of the boundary layer, to a value of 0 forthe outer wake region of the boundary layer, and for free shear layers. The model coefficients, Փ={
σ_{k}, σ_{ω}, γ, β_{ω}
} are obtained by means of the blending function:
Two functions are needed; and the eddy viscosity relation for the SST model is given by:
 2.3 Compressibility Corrections
Inspection of the turbulence kinetic energy equation also indicates that the Favreaveraged dissipation rate is given by
where,
Thus, the compressible turbulence dissipation rate can logically be written in terms of the fluctuating vorticity, and the divergence of the fluctuating velocity. At high Reynolds number, these components are presumably uncorrelated. The quantity
ε_{s}
is known as the solenoidal dissipation, while
ε_{d}
is known as the dilatation dissipation. Clearly, the lattercontribution is present only for compressible flows.
Based on observations from some older Direct Numerical Simulation (DNS), Sarkar et al.
[6
,
7]
postulate that the dilatation dissipation should be a function of the turbulence Mach number,
M_{t}
, defined by
where, a is the speed of sound. They further argue that the k and ε equations should be replaced by
where,
C_{ε2}
is a closure coefficient. Only the dissipation terms are shown explicitly in Equations (21) and (22), since no changes occur in any other terms. The equation for
ε_{s}
is unaffected by compressibility. The dilatation dissipation is further assumed to be proportional to
ε_{s}
, so that
where,
ξ
* is a closure coefficient, and
F(M_{t})
is a prescribedfunction of
M_{t}
. Building upon the Sarkar formulation anddimensional analysis, Wilcox has postulated a similar modelthat enjoys an important advantage for wallbounded flows
[9]
. The Sarkar and Wilcox formulations differ in the value of
ξ
*, and the functional form of
F(M_{t})
. The values of
ξ
* and
F(M_{t})
for the Sarkar and Wilcox models are:
where,
H
(
x
) is the Heaviside step function.
The pressuredilatation is large, in flows with a large ratio of turbulenceenergy production to dissipation. Therefore, Sarkar proposes that the pressure dilatation can be approximated as
where,
M_{t}
is the turbulence Mach number, and the closure coefficients
α
_{2}
and
α
_{3}
are given by
The model has been used for a range of compressibleflow applications, including mixing layers, and attached boundary layers.
To implement the Sarkar modification in the kω model, the formal change of variables given by
ε_{s} = β*ωk
is made. If, for simplicity,
β*
is a pure constant, this equation changes,as follows;
Consequently, a compressibility term must appear in the ω equation, as well as in the
k
equation. Sarkar and Wilcox compressibility modifications correspond to letting closurecoefficients
β
and
β*
in the kω model vary with
M_{t}
. In terms of
ξ
* and the compressibility function
F(M_{t})
,
β
and
β*
become:
where,
β_{0}*f_{β*}
and
β_{0}f_{β}
are the corresponding incompressible values of
ξ
* and
β
[10]
.
 2.4 Discretization Method
The concept of total variation diminishing (TVD), the first of the upwind schemes applied in this study, is that the total variation of any of the physical properties does not increase in time. Temporal discretization is obtained, using a second order dual time stepping integration. The spatial convective terms are discretized with the thirdorder upwind TVD scheme
[6]
.
In order to avoid such nonphysical oscillation, the AUSMPW+ scheme was chosen for the inviscid flux calculation. In previous research, the AUSMPW+ scheme has been known to be more stable, and at the same time less dissipative in supersonic flow, than the Roe scheme. The AUSMPW+ scheme is a modified version of the AUSM (Advection Upstream Splitting Method) family of schemes. The AUSM concept is to use different splitting for the convective fluxes, with each splitting being some function of an intuitively defined interface Mach number. The AUSMPW scheme with pressurebased weight function wasintroduced to overcome the carbuncle phenomenon, and the overshoot problems behind a strong shock, in the AUSM or AUSMD scheme. The AUSMPW+ scheme is an improved and simplified version of the AUSMPW scheme, with a new numerical speed of sound introduced
[21]
.
 2.5 Numerical scheme
The conservation equations for moderate and high Mach number flows are wellcoupled, and standard numerical techniques perform adequately. In regions of low Mach number flows, however, the energy and momentum equations are practically decoupled, and the system of conservation equations becomes stiff. To overcome this problem, a twostep dual timeintegration procedure designed for all Mach number flows was applied. First, a rescaled pressure term is used in the momentum equation,
Schematic of the geometry, and a magnified view of the representative meshing around the transonic bump [12]
in order to circumvent the singular behavior of pressure at low Mach numbers. Second, a dual timestepping integration procedure is established.
The pseudotime derivative may be chosen, in order to optimize the convergence of the inner iterations, by using an appropriate preconditioning matrix that is tuned to rescale the eigenvalues to render the same order of magnitude, so as to maximize convergence. To unify the conserved flux variables, a pseudotime derivative of the form Г
∂Z/∂τ
can be added to the conservation equation. Since the pseudotime derivative term disappears upon convergence, a certain amount of liberty can be taken, in choosing the variable Z. In this study, a pressure,
p'
, is introduced as the pseudotime derivative term in the continuity equation
[22]
.
A multiblock feature using an MPI library was used to speed up the calculation
[22]
.
3. Results
 3.1 Arcbump
The bump geometry is one of the most broadly used configurations for validating turbulence models in shock/boundary layer interaction in transonic flows. The experiment of Inger and Gendt
[11
,
12]
considers a channel with a flat floor wall, and an upper surface with a bump, which has a circular arc of 580 mm radius of curvature and 20mmheight. The inlet flow is subsonic, and there are no incoming standing waves. A schematic of the bump geometry and the computational domain are shown in
Fig. 1
.
About 29,000 grid points are used; the grid distribution is dense near the wall, in order to accurately capture the boundary layer. The numerical boundary conditions are: stagnation pressure of 140
kpa
, stagnation temperature of 310
^{o}
K
, Mach number of
M_{∞}
=0.73, and exit pressure ratio of
p/p_{∞}
=0.68.
Figure 2
shows the Mach number and pressure contours on the arc bump. The velocity accelerates from transonic (
M_{∞}
=0.73) to supersonic (about
M
=1.3) behind the leading edge of the arc bump (x=0), due to variation of the flow path area. The normal shock is located at about 70% of the arc bump, with the Mach number being about 1.3. The separation region, which has a low Mach number of about 0.1, occurs behind the normal shock.
Figure 3
shows the variation of the pressure coefficient
C_{p}
along the upper wall, for the experimental data of Inger and Gendt
[11]
, and the kεand kωmodels with and without the Sarkar correction. The predictions upstream of the normal shock are in good agreement with the experimental data, for all turbulence test models. In the separation region, however, several of the test models vary somewhat from the experimental data.
The kω SST model predicts most closely the normal shock location, but the pressure recovers slowly after the shockwave. The compressibility effect, the Sarkar model, does not much effect the normal shock location and overall tendency of
C_{p}
. Note particularly that the kω SST with the Sarkar model predicts the normal shock location better than the low Reynolds kε model. The kε model without any compressibility correction predicts the normal shock location relatively upstream, but shows better pressure recovery after the shock.
Figure 4
shows a comparison of two different compressibility correction models, the Sarkar and Wilcox models. The Sarkar model considers only the local turbulent Mach number, while the Wilcox model uses both the
Mach number and pressure contours of the arc bump configuration
Variation of C_{p} along the channel
standard turbulent Mach number, and the local turbulent Mach number, to consider the dilatation dissipation. The result for the low Reynolds kε model using the Wilcox compressibility correction model presents better shock location and pressure recovery, than others.
Based on these results, the low Reynolds kε model with Wilcox compressibility correction is seen to most accurately predict the experimental data.
 3.2 Transonicdiffuser
Twodimensional simulation of the convergent / divergent diffuser described experimentally by Borger
[13]
has been conducted. The bottom wall of the diffuser is flat, but the upper wall is given by
where,
The various constants for the top wall are given in
Table 1
. The throat height is
h_{th}
=4.4cm.
Figure 5
shows the geometry and grid distribution of the 100×70 grid points used in this study. At the inflow boundary,
Comparison of two compressibility models.
the stagnation pressure is 135
kpa
, and the stagnation temperature is 277.8
^{o}
K
[13
,
14]
.
The pressure ratio, defined as the static pressure at the outlet, divided by the inlet total pressure, characterizes the diffuser flow. The weak shock and the strong shock in the diffuser are simulated. The pressure ratios of the weak shock and the strong shock are 0.82 and 0.72, respectively.
Figure 6
shows the flow structure of the diffuser for strongshock and weakshock conditions. In the weakshock
Constants for channel height[15]
Constants for channel height [15]
Schematic of the geometry, and grid distribution of the transonicdiffuser [14].
Mach number contours in a diffuser, for weakand strongshock conditions.
Comparison of pressure ratios along diffuser top and bottom walls, for weakshock condition.
condition, the normal shock is located at around
X/h_{s}
=1.5, with a Mach number of 1.2, and the separation region appears behind the normal shock, on the top side of the diffuser. On the other hand, in the strongshock condition, the normal shock is located at around
X/h_{s}
=2.0, with a Mach number of 1.4, and a larger separation region is observed
Figure 7
shows a comparison of the pressure ratio along the top and bottom walls of the diffuser, for the weakshock condition. Both the low Reynolds number kε model, and the kω SST model, give accurate results. The differences in the results using the Sarkar and Wilcox compressibility models are so small, that the numerical data using the Wilcoxcompressibility model is not presented.
To investigate the effects of the size of recirculation and shock strength, the strongshock condition has been investigated. In the strongshock condition, the separation region is larger, and the shock is much stronger, than in the weakshock condition.
Figure 8
shows a comparison of the pressure ratio for the strongshock condition. The kω SST model predicts the normal shock location better than the low Reynolds kε model does. The low Reynolds kε model with the Sarkar correction predicts the normal shock location more accurately, but not as well as the kω model.
On the bottom wall, the separation region is smaller, than on the top side of the diffuser. The results on the bottom wall behind the normal shock are more accurate, than those on the top wall.
These results show that both twoequation turbulence models have accurate results in the small separation region, but for strong shock condition, the kω SST model predicts the shock position better than the low Reynolds kε model. The compressibility effects on the kω SST model are negligible.
 3.3 Supersonic jet impingement
To investigate the compressibility effect, a numerical study of a supersonic jet impinging on an axisymmetric cone has been conducted.
The designed Mach number of the nozzle exit is 2.0, and the mean ratios of nozzle exit pressure to atmospheric pressure (PR) are 1.2 and 2.27. The inlet conditions at PR=1.2 and 2.27 are a total pressure of 921kPa, and 1,742kPa,
Pressure ratio variation on the top and bottom walls, at strong shock condition.
Schematics and computational domain of the supersonic nozzle and axisymmetric cone [15]
respectively, and the total temperature is 300K for both cases
[15]
. The apex angle of the cone is 90˚, and the distancebetween the nozzle exit and the apex point is
Z_{n}
of 13mm, and
Z_{n}/D_{e}
=1.0(
Fig. 9
(a)).
Figure 9
(b) shows the axisymmetric computational domain with a grid system. The total number of grid points is about 15,520. Three blocks and six processors are applied, to improve the calculation efficiency.
Figure 10
shows the density gradient magnitude, and the Mach number contours. The maximum Mach numbers are about 2.6, in the case of PR=1.2, and about 3.2, in the case of PR=2.27.
The impinging jet flow generates an expansion wave at the nozzle exit, jet boundary, barrel shock, and Mach disk. Due to the impingement on the apex of the cone, the structure of the jet flow becomes more complex. In PR=1.2, the subsonic zone is near the apex of the cone, because the Mach disk is in front of the apex of the cone. The Mach disk rapidly reduces the jet velocity to subsonic. The jet flow through the Mach disk attaches to the surface of the cone, and is recompressed. The pressure recovery at the second compression is more gradual than PR=2.27, as shown in
Fig. 10
. This means that the degree of the compressibility and the interaction of the shock wave with a turbulent boundary is weaker, than the case of PR=2.27. In PR=2.27, the jet flow core impinges to the apex of the cone, and a stagnation point occurs. The oblique shock at the cone apex closes to the cone wall, and the pressure at the second compression increases more steeply.
The experimental data and numerical results without compressibility correction are comparable. The compressibility effects are negligible, because 1) the flow separation region at the cone surface is not large, so that the interaction between turbulent boundary and jet flow is not active, and 2) the compressible wave and perturbations due to the impinging on the cone propagate out to the atmosphere, but do not bounce back.
Density gradient magnitude (left), and the Mach number (right) contours
Pressure ratio along the edge wall of the cone
 3.4 Supersonic diffuser
To determine the accuracy of the two turbulence models and the compressibility models in the supersonic regime, the calculation of a supersonic exhaust diffuser configuration was conducted.
Fig. 12
shows a schematic view, and the computational domain of the supersonic diffuser.
The dimensions of the model are diffuser length L of 260mm, diffuser diameter D of 21 mm, ratio of length to diameter of diffuser L/D of 12.38, area ratio of diffuser to rocket motor nozzle throat
A_{d}/A_{t}
of 56.25, and area ratio of diffuser exit to rocket motor nozzle throat
A_{e}/A_{t}
of 35.20. The computational domain consists of three blocks, and each block grid number is 115 x 50, 79 x 30, and 206 x 79, respectively. The boundary condition has the same condition as the experiment. At the inlet boundary, the stagnation pressure is 50 bar (5
MPa
), the stagnation temperature is 300
^{o}
K, and the pressure of the outlet boundary condition is 1 atm
[16
,
17]
.
Figure 13
shows the Mach number contours in the supersonic diffuser. The maximum Mach number of about 6.7 appears in the first diamond shock region. A smallsupersonic pocket occurs at the axis of the diffuser, and moves periodically downstream and upstream. The reason that the shock train accompanying pressure oscillation occurs in the diffuser is that the mass flux and momentum of the exhaust jet are not enough to block the pressure
Schematic and computational domain of the supersonic exhaust diffuser
Mach number contours in supersonic diffuser (△t=5ms)
Comparison of pressure variation along the diffuser wall, according to the compressibility model
oscillation produced by the acousticwave oscillation, in the subsonic region behind the terminal shock. The period of oscillation may relate to the acoustic mode in the subsonic region, after the second shock diamond. Thus, the vacuumchamber pressure stays constant, but the pressure along the wall oscillates periodically, with a certain amplitude, as shown in
Fig. 13
. If the rocket motor pressure decreases further, the shock train becomes weak.
Figure 14
shows a comparison of the pressure along the wall of the diffuser, according to the compressible correction models. The pressure rises behind the impinging point of the jet on the diffuser wall, after the pressure decreases in the expansion region. Afterward, the pressure increases again during the next compression wave; and then it finally rises to atmospheric pressure, at the exit of the diffuser. The results of the kω SST model are very different from the experimental data, since the kω SST model does not predict the oscillatory flow structure, or the strong unsteady motion. Using the compressibility model, the Sarkar model produces better results, but it does not compare well with the experimental data. The low Reynolds number kε model with the Sarkar model is in fairly good agreement with the experimental data. Therefore, in the supersonic regime, the experimental data and the numerical results, from the low Reynolds kε model with the Sarkar model, appear to reveal a discrepancy; but the unsteady flow motion in the diffuser is hidden, in the background of the instantaneous data.
Figure 14
(b) shows a comparison of the pressure along the wall, using the Sarkar and Wilcox models. The kω SST models are unreliable with both compressibility models, because the kω SST model is weak in strong unsteady flow simulation. On the other hand, the low Reynolds number kε model has more accurate results, for this particular unsteady flow.
4. Conclusions
To evaluate numerical modeling of compressibility effects, two twoequation turbulence models, the low Reynolds number kε model and the kω SST model, with two compressibility models proposed by Sarkar and Wilcox, are applied to simulate four different transonic or supersonic flows. The numerical results are evaluated against experimental data.
For the first validation, a transonic arc bump, the kω SST model predicts the normal shock location fairly well, but the pressure after the shock recovers slowly, and the compressibility effects are negligible. The low Reynolds kε model with the Wilcox compressibility effect presents better accuracy.
For the second validation case, a transonic diffuser, both the weak shock and strong shock cases are evaluated. Both twoequation turbulence models reasonably predict the flow properties for the weak shock condition, but the kω SST model offers more accurate results for the strong shock condition. The compressibility effects on the kω SST model are negligible.
For the third validation case, a supersonic impinging jet, supersonic jets of pressure ratio (PR) 1.2 and 2.27 are investigated. Both twoequation turbulence models have similar wall pressures, with reasonable accuracy against experimental data, because the flow separation size is small, and the compressible wave and perturbation due to the impinging on the cone smear out to the atmosphere.
Finally, an unsteady supersonic diffuser was evaluated for the test of the compressibility correction models. The low Reynolds kε equation with Wilcox compressibility model provides the most accurate results for this particular unsteady flow.
In summary, compressibility effects on the turbulence model are very important in supersonic flow with large flow separation, because of the strong interaction between shock and boundary layer. The compressibility correction model has a stronger influence on the kε model, than on the kω SST model.
Acknowledgements
This work was supported by Defense Acquisition Program Administration and Agency for Defense Development under the contract UD110093CD.
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