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Fig. 7 Temperature contours in Kelvin for the ramp injector case at x = 0.02 m using a 8-million-node mesh. comparison between the steady-state temperature contours down- stream of the point of injection is shown in Fig. 7 using a mesh comprising 8 million nodes. The temperature profiles show that the

Figure 7 Temperature contours in Kelvin for the ramp injector case at x = 0.02 m using a 8-million-node mesh. comparison between the steady-state temperature contours down- stream of the point of injection is shown in Fig. 7 using a mesh comprising 8 million nodes. The temperature profiles show that the

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where f and g are the fluxes along x and y, respectively. To do so, first rewrite the discretization equation [Eq. (12)] in the following form: where 3, is a parameter related to the splitting of the cross-derivative terms among the several dimensions in order to obtain monotonicity. The value that needs to be given to f,,, to achieve monotonicity will be outlined in a later section. Note that, independent of the value given to By, the latter equation will yield exactly the same solution as Eq. (12). Now, we can split the cross-derivative terms function of 3, as follows:
where f and g are the fluxes along x and y, respectively. To do so, first rewrite the discretization equation [Eq. (12)] in the following form: where 3, is a parameter related to the splitting of the cross-derivative terms among the several dimensions in order to obtain monotonicity. The value that needs to be given to f,,, to achieve monotonicity will be outlined in a later section. Note that, independent of the value given to By, the latter equation will yield exactly the same solution as Eq. (12). Now, we can split the cross-derivative terms function of 3, as follows:
Following the same steps as shown previously, it can be easily shown that the modified equation associated with the latter collapses to the original differential equation [Eq. (1)] as long as a/ Ax and b/Ay are within the same order of magnitude. It is emphasized that Eq. (12) is a second-order-accurate dis- cretization of a hyperbolic equation with constant wave speeds. When applied to a hyperbolic system where the wave speeds are not constant (such as the Euler equations for instance), the second-order accuracy is not guaranteed. Nonetheless, this is not a source of concern, as the goal here is not to obtain a second-order-accurate scheme (as in [28], for instance) but rather a first-order-accurate genuinely multidimensional extension of the Roe scheme with minimal truncation error. How this can be obtained from Eq. (12) is outlined next.
Following the same steps as shown previously, it can be easily shown that the modified equation associated with the latter collapses to the original differential equation [Eq. (1)] as long as a/ Ax and b/Ay are within the same order of magnitude. It is emphasized that Eq. (12) is a second-order-accurate dis- cretization of a hyperbolic equation with constant wave speeds. When applied to a hyperbolic system where the wave speeds are not constant (such as the Euler equations for instance), the second-order accuracy is not guaranteed. Nonetheless, this is not a source of concern, as the goal here is not to obtain a second-order-accurate scheme (as in [28], for instance) but rather a first-order-accurate genuinely multidimensional extension of the Roe scheme with minimal truncation error. How this can be obtained from Eq. (12) is outlined next.
Thus far, the discretization equation has not been modified. Indeed, should the fluxes outlined in Eqs. (17) and (18) be substituted in Eq. (13), one would obtain precisely the discretization equation outlined in Eq. (12). After substituting in Eq. (14), and comparing the resulting equation to Eq. (13), it follows that the flux functions at the interfaces correspond to
Thus far, the discretization equation has not been modified. Indeed, should the fluxes outlined in Eqs. (17) and (18) be substituted in Eq. (13), one would obtain precisely the discretization equation outlined in Eq. (12). After substituting in Eq. (14), and comparing the resulting equation to Eq. (13), it follows that the flux functions at the interfaces correspond to
After substituting At;,)/. and Ati in Eqs. (17) and (18), respectively, the fluxes become In the fluxes outlined in Eqs. (24) and (25), the first-derivative and the second-derivative terms (the first two terms on the right-hand side) respect the rule of the positive coefficients and lead to a monotonic flux function. However, the cross-derivative terms (all the term functions of #) do not obey the rule of the positive coefficients. Further, it can be demonstrated that no value given to Z would yield a discretization equation with all-positive coefficients. One way that the method could be made monotonicity preserving is by limiting the cross-derivative terms such that the resulting fluxes obey the total- variation-diminishing (TVD) condition. Numerical experiments indicate, however, that such a strategy results in erratic limiter behavior often prevents a converged solution. An alternative novel method is here proposed that yields essentially monotone solutions while not being subject to convergence hangs. The method consists of defining / such that the magnitude of the cross-derivative coefficient is not superior to the magnitude of the second-derivative coefficient. This results in solutions that are essentially monotonic because the spurious oscillations that may originate from the cross-derivative terms are damped by the dissipation introduced by the second- order terms. It is noted that the latter two equations essentially define the grid spacings Ax and Ay for a given time step At = Afj44/2 = Atj+1/2- Should the grid spacing be specified differently, the resulting discretization equation would still yield a solution that is monotonic, but the truncation error would be more significant.
After substituting At;,)/. and Ati in Eqs. (17) and (18), respectively, the fluxes become In the fluxes outlined in Eqs. (24) and (25), the first-derivative and the second-derivative terms (the first two terms on the right-hand side) respect the rule of the positive coefficients and lead to a monotonic flux function. However, the cross-derivative terms (all the term functions of #) do not obey the rule of the positive coefficients. Further, it can be demonstrated that no value given to Z would yield a discretization equation with all-positive coefficients. One way that the method could be made monotonicity preserving is by limiting the cross-derivative terms such that the resulting fluxes obey the total- variation-diminishing (TVD) condition. Numerical experiments indicate, however, that such a strategy results in erratic limiter behavior often prevents a converged solution. An alternative novel method is here proposed that yields essentially monotone solutions while not being subject to convergence hangs. The method consists of defining / such that the magnitude of the cross-derivative coefficient is not superior to the magnitude of the second-derivative coefficient. This results in solutions that are essentially monotonic because the spurious oscillations that may originate from the cross-derivative terms are damped by the dissipation introduced by the second- order terms. It is noted that the latter two equations essentially define the grid spacings Ax and Ay for a given time step At = Afj44/2 = Atj+1/2- Should the grid spacing be specified differently, the resulting discretization equation would still yield a solution that is monotonic, but the truncation error would be more significant.
Then, substitute ,,, in the fluxes outlined in Eqs. (24) and (25), and simplify Although not monotonicity preserving in the strict sense following Harten’s definition, numerical ex periments indicate t hat the latter flux functions yield a solution that is essentially free of spurious oscillations. Several comparisons posed method and the dimensiona! scalar conservation law, not only -splitting strategy w were performed between the pro- hen solving a for a constant-coefficient test case but also for several other test cases in which the coefficients were not constant and given an arbitrary spatial distribution including discontinuities. It is found that t he proposed method yields a re- solution being significantly superior to the one obtained with dimensionally split first-order upwinded stencils. In fact, despite being first-order accurate, the present approach is found to yield a resolution that is essentially the same as the one of dimensionally split second-order TVD stencils whenever the waves propagate obliquely to the grid lines. Although the cross-derivative terms may introduce small aphysical oscillations near sharp gradients, such oscillations are barely discernible and are quickly damped by the second- derivative terms.
Then, substitute ,,, in the fluxes outlined in Eqs. (24) and (25), and simplify Although not monotonicity preserving in the strict sense following Harten’s definition, numerical ex periments indicate t hat the latter flux functions yield a solution that is essentially free of spurious oscillations. Several comparisons posed method and the dimensiona! scalar conservation law, not only -splitting strategy w were performed between the pro- hen solving a for a constant-coefficient test case but also for several other test cases in which the coefficients were not constant and given an arbitrary spatial distribution including discontinuities. It is found that t he proposed method yields a re- solution being significantly superior to the one obtained with dimensionally split first-order upwinded stencils. In fact, despite being first-order accurate, the present approach is found to yield a resolution that is essentially the same as the one of dimensionally split second-order TVD stencils whenever the waves propagate obliquely to the grid lines. Although the cross-derivative terms may introduce small aphysical oscillations near sharp gradients, such oscillations are barely discernible and are quickly damped by the second- derivative terms.
The improvement in resolution obtained with the proposed method is not limited to contact surfaces and shear waves, but it is apparent through shock waves and expansion fans. For instance, consider air entering a channel at a Mach number of two, a pressure of 10,200 Pa, and a temperature of 300 K. When the flow interacts with the wavy Several test cases are now considered to assess the gain in resolution of the proposed genuinely multidimensional method over the dimensionally split Roe solver [30] and the dimensionally split Yee—Roe TVD scheme [31]. All cases presented herein consist of the steady-state solution of either the Euler equations or Navier-Stokes equations on a structured mesh using generalized curvilinear coordinates and a perfect gas model with the specific heat ratio set to 1.4 and the gas constant set to 286 J/kg - K. Additional numerical
The improvement in resolution obtained with the proposed method is not limited to contact surfaces and shear waves, but it is apparent through shock waves and expansion fans. For instance, consider air entering a channel at a Mach number of two, a pressure of 10,200 Pa, and a temperature of 300 K. When the flow interacts with the wavy Several test cases are now considered to assess the gain in resolution of the proposed genuinely multidimensional method over the dimensionally split Roe solver [30] and the dimensionally split Yee—Roe TVD scheme [31]. All cases presented herein consist of the steady-state solution of either the Euler equations or Navier-Stokes equations on a structured mesh using generalized curvilinear coordinates and a perfect gas model with the specific heat ratio set to 1.4 and the gas constant set to 286 J/kg - K. Additional numerical
fable 1 Percent error on the density and estimation of the order of accuracy using various schemes for the sine wall test case *The exact solution is obtained using the Yee-Roe TVD scheme on a 3201 x 801 node mesh. >The reference surface area S,o¢ is set to 1 m, and the reference density p;cr is set to 0.119 kg/m?. ‘The order of accuracy is estimated using Richardson extrapolation as p = In(e,/e,)/In(Ax,/Ax,) with € as the erro and the subscripts c and f referring to the coarse and fine meshes, respectively.
fable 1 Percent error on the density and estimation of the order of accuracy using various schemes for the sine wall test case *The exact solution is obtained using the Yee-Roe TVD scheme on a 3201 x 801 node mesh. >The reference surface area S,o¢ is set to 1 m, and the reference density p;cr is set to 0.119 kg/m?. ‘The order of accuracy is estimated using Richardson extrapolation as p = In(e,/e,)/In(Ax,/Ax,) with € as the erro and the subscripts c and f referring to the coarse and fine meshes, respectively.
Table 3 Number of iterations to reach convergence for the sine wall test case using a 301 x 76 mesh and block-implicit approximate- factorization pseudotime stepping: ‘Convergence is reached when the residual diminishes below a certain user-defined constant, which is fixed to the same value for all schemes and for all CFL numbers; this entails approximately seven orders of magnitude of convergence.
Table 3 Number of iterations to reach convergence for the sine wall test case using a 301 x 76 mesh and block-implicit approximate- factorization pseudotime stepping: ‘Convergence is reached when the residual diminishes below a certain user-defined constant, which is fixed to the same value for all schemes and for all CFL numbers; this entails approximately seven orders of magnitude of convergence.
Table 2 CPU time per iteration (in seconds) for the sine wall test case using a 301 x 76 mesh
Table 2 CPU time per iteration (in seconds) for the sine wall test case using a 301 x 76 mesh
Fig.3 Plot of the 33 x 33 mesh used for the Mach 3 blunt body test case. Another test case that is commonly used to assess the capabilities of a numerical method in capturing shock waves is the simulation of
Fig.3 Plot of the 33 x 33 mesh used for the Mach 3 blunt body test case. Another test case that is commonly used to assess the capabilities of a numerical method in capturing shock waves is the simulation of
A) TAU SLU al SUMCINY [aU] Wy ELUPUSCE LEP SUWULUCE SUMCIIC QR) OCLUMUSULUCE £ VE? SULCIIC 4 Pressure coefficient contours for the blunt body case with ACp = 0.05. Figure 4a is reprinted from [26] with permission from Elsevier. It is noted, however, that the proposed method does exhibit somewhat lower resolution than the dimensionally split approach on the basis of pressure profiles on the axis of symmetry (see Fig. 5a), where the shock can be seen to be slightly more smeared. This is not a particular source of concern, however, because 1) this becomes less significant as the grid is refined (see Fig. 5b), and 2) this is limited to small regions and does not significantly affect the solution over the rest of the domain. Indeed, for many practical problems, the size of
A) TAU SLU al SUMCINY [aU] Wy ELUPUSCE LEP SUWULUCE SUMCIIC QR) OCLUMUSULUCE £ VE? SULCIIC 4 Pressure coefficient contours for the blunt body case with ACp = 0.05. Figure 4a is reprinted from [26] with permission from Elsevier. It is noted, however, that the proposed method does exhibit somewhat lower resolution than the dimensionally split approach on the basis of pressure profiles on the axis of symmetry (see Fig. 5a), where the shock can be seen to be slightly more smeared. This is not a particular source of concern, however, because 1) this becomes less significant as the grid is refined (see Fig. 5b), and 2) this is limited to small regions and does not significantly affect the solution over the rest of the domain. Indeed, for many practical problems, the size of
Fig. 6 Schematic of the ramp injector case with all dimensions in millimeters. Through the aforementioned test cases, it was demonstrated how the addition of cross-derivative terms to the Roe flux function can yield significant enhancements in resolution when solving expansion fans and shock waves, as well as shear waves. Another flow feature that necessitates investigation is the streamwise vortex occurring in some three-dimensional flowfields. One example of a streamwise vortex that is particularly difficult to simulate is the one generated by ramp injectors in scramjet inlets or combustors, with hydrogen or some hydrocarbon fuel being the injectant (see, for instance, [36] or [37]). To make the problem more easily reproducible, the injected fuel is here replaced by air with the inflow conditions being set as in Fig. 6. A surface of symmetry is imposed at z + 28.6 mm whereas the viscosity is fixed to 2 x 10-> kg/ms and the thermal conductivity to 0.03 W/mK. Due to the incoming Mach number being in the hypersonic range, a strong bow shock forms above the injector. This leads to a large pressure difference between the top and side surfaces of the ramp, effectively inducing some streamwise vortices along the injector side surfaces. To minimize the impact of physical dissipation on the streamwise vortices, no turbulence model is added to the governing equations (the governing equations are hence limited to the mass conservation equation, the Navier-Stokes equations, and the total energy equation including thermal diffusion and viscous dissipation). Because of the low physical dissipation, the vortex retains its strength and continuously distorts and stretches the interface between the injected air and the incoming air throughout the mixing region. Such a flowfield is particularly difficult to capture due to the sharp interface between the injectant and the incoming flow moving rapidly in various directions and traveling obliquely to the grid lines, hence leading to a significant amount of numerical dis- sipation within the vortex. To assess the amount of dissipation introduced by the numerical method within the mixing region, a
Fig. 6 Schematic of the ramp injector case with all dimensions in millimeters. Through the aforementioned test cases, it was demonstrated how the addition of cross-derivative terms to the Roe flux function can yield significant enhancements in resolution when solving expansion fans and shock waves, as well as shear waves. Another flow feature that necessitates investigation is the streamwise vortex occurring in some three-dimensional flowfields. One example of a streamwise vortex that is particularly difficult to simulate is the one generated by ramp injectors in scramjet inlets or combustors, with hydrogen or some hydrocarbon fuel being the injectant (see, for instance, [36] or [37]). To make the problem more easily reproducible, the injected fuel is here replaced by air with the inflow conditions being set as in Fig. 6. A surface of symmetry is imposed at z + 28.6 mm whereas the viscosity is fixed to 2 x 10-> kg/ms and the thermal conductivity to 0.03 W/mK. Due to the incoming Mach number being in the hypersonic range, a strong bow shock forms above the injector. This leads to a large pressure difference between the top and side surfaces of the ramp, effectively inducing some streamwise vortices along the injector side surfaces. To minimize the impact of physical dissipation on the streamwise vortices, no turbulence model is added to the governing equations (the governing equations are hence limited to the mass conservation equation, the Navier-Stokes equations, and the total energy equation including thermal diffusion and viscous dissipation). Because of the low physical dissipation, the vortex retains its strength and continuously distorts and stretches the interface between the injected air and the incoming air throughout the mixing region. Such a flowfield is particularly difficult to capture due to the sharp interface between the injectant and the incoming flow moving rapidly in various directions and traveling obliquely to the grid lines, hence leading to a significant amount of numerical dis- sipation within the vortex. To assess the amount of dissipation introduced by the numerical method within the mixing region, a
Table 4 Comparison between the proposed method and the dimensionally split first order approach on the basis of stagnation pressure difference for the ramp injector test case at x = 0.2 m* *The stagnation pressure difference AP, corresponds to the difference between the freestream stagnation pressure and the local stagnation pressure determined from perfect gas relationships with the specific heat ratio set to 7/5.
Table 4 Comparison between the proposed method and the dimensionally split first order approach on the basis of stagnation pressure difference for the ramp injector test case at x = 0.2 m* *The stagnation pressure difference AP, corresponds to the difference between the freestream stagnation pressure and the local stagnation pressure determined from perfect gas relationships with the specific heat ratio set to 7/5.
Table 5 Comparison between the proposed method and the dimensionally split schemes on the basis of drag force for the subsonic channel test case
Table 5 Comparison between the proposed method and the dimensionally split schemes on the basis of drag force for the subsonic channel test case
braer Noe solver Db) second-order 1 VU scheme Cc) Froposed first-order scheme Fig. 8 Mach number contours for the subsonic case with AM = 0.05 using a 183 x 183 mesh.
braer Noe solver Db) second-order 1 VU scheme Cc) Froposed first-order scheme Fig. 8 Mach number contours for the subsonic case with AM = 0.05 using a 183 x 183 mesh.
Fig.9 Skin friction coefficient for the flat plate case.
Fig.9 Skin friction coefficient for the flat plate case.
where Fj41/2, Gj41/2, and H;+1/2 denote the flux functions at the interfaces perpendicular to the x, y, and z axes, respectively. Then, following the same steps as in Secs. II-V, it can be shown that the proposed flux functions at the interfaces correspond to
where Fj41/2, Gj41/2, and H;+1/2 denote the flux functions at the interfaces perpendicular to the x, y, and z axes, respectively. Then, following the same steps as in Secs. II-V, it can be shown that the proposed flux functions at the interfaces correspond to
In the latter, A, B, and C correspond to the flux Jacobians, whereas |A|, |B|, and |C| correspond to the Roe matrices along x, y, and z, respectively.
In the latter, A, B, and C correspond to the flux Jacobians, whereas |A|, |B|, and |C| correspond to the Roe matrices along x, y, and z, respectively.
Related topics:
Numerical SimulationsAerodynamicsCompressible Fluid DynamicsComputational MathematicsShock WavesComputational Fluid Dynamics (CFD) modelling and simulationSupersonic flow
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