A Coupled Two-Domain Approach for Transpiration Cooling

The flow in the hot gas domain is assumed to be turbulent. Therefore, we apply a RANS model that formally can be written asIn our numerical investigations we will consider both the flow in a subsonic thrust chamber at moderate temperatures and a supersonic nozzle at high temperatures. These investigations are motivated by experiments performed in the wind tunnel  [19, 25] at University Stuttgart and the detonation-based facility for rocket-engine-like flows  [9, 21, 32] at the Shock Wave Laboratory at RWTH Aachen. Therefore, the system (1) of RANS equations will differ for the two scenarios.

Scenario 1: Subsonic thrust chamber. In this scenario the gas is treated as a thermally and calorically perfect gas. The turbulence is modeled by the Wilcox k-\(\omega \) model  [29]. Thus, the system (1) consists of the continuity equation, the momentum equation and the energy equation for density \(\rho \), momentum \(\rho \mathbf {v}\) and total energy E as well as two equations for the turbulence kinetic energy k and the specific dissipation rate \(\omega \) with \(\mathbf {v}\) the fluid velocity. The vector of conserved quantities is determined bywhere the density is Reynolds-averaged and all other quantities are Favre-averaged. The system is closed by an equation of state for a thermally perfect gas. For further details, in particular on the modeling of the Reynolds stress tensor as well as the mean and the turbulent heat flux, we refer to previous work  [4, 6, 17]. The following additional quantities enter the model: the specific heat capacity at constant pressure \(c_p\), the dynamic viscosity \(\mu \) and the turbulent viscosity \(\mu _t\), the Prandtl number Pr and the turbulent Prandtl number \(Pr_t\), the specific gas constant R and the isentropic exponent \(\gamma \).

Scenario 2: Supersonic nozzle. In this scenario the hot gas model has to account for a thermally perfect but calorically imperfect gas mixture of \(N_s\) non-reacting species. For this purpose, the RANS equations (1) are extended by additional \(N_s\) species equations for the partial densities \(\rho _\alpha \), i.e., the vector of conserved quantities is now determined byHere the turbulent quantities are modeled by Menter’s SST k-\(\omega \) model  [22]. The system (1) is closed by an equation of state for a thermally perfect gas mixture. For details we refer to  [3, 30].

The system (1) is complemented by suitable boundary conditions for the two scenarios:

Scenario 1: Subsonic thrust chamber. At the inflow boundary \(\Gamma _{\text {I}}\) the Mach number, the density and the temperature are prescribed by \(M =M_\infty \), \(\rho =\rho _\infty \) and \(T=T_\infty \). The turbulence kinetic energy is defined by the turbulent intensity \(Tu_\infty = \sqrt{2/3 \, k_\infty } / |\mathbf {v}_{\infty }|\), where \(\mathbf {v}_{\infty }\) is the inflow velocity vector. The turbulent dissipation rate \(\omega _{\infty }\) is computed by \(\rho _{\infty } k_{\infty }/\mu _t\), where the ratio of the turbulent dynamic viscosity to the laminar dynamic viscosity \((\mu _t / \mu )_\infty \) is used in combination with a viscosity curve fit model. At the outflow boundary \(\Gamma _{\text {O}}\), only the pressure is prescribed by setting \(p = p_\infty \). For the walls \(\Gamma _{\text {W,HG,iso}}\) of the channel we use isothermal boundary conditions motivated by the thermal wall behavior observed in the experiments in  [25]. For the walls \(\Gamma _{\text {W,HG,ad}}\) downstream of the porous sample adiabatic boundary conditions account for the changing wall temperature due to the cooling.

Scenario 2: Supersonic nozzle. At the nozzle entry \(\Gamma _{\text {I}}\) we additionally set the mass fractions \(X_{\alpha ,\infty }=\rho _{\alpha ,\infty }/\rho _\infty \) of each species \(\alpha \). At the nozzle exit \(\Gamma _{\text {O}}\) only the outflow pressure is prescribed by setting \(p = p_{\text {out}}\). Due to the short test duration of a few milliseconds in the corresponding experiments  [9, 21, 32], there is no wall temperature increase. Hence, all nozzle walls are modeled as isothermal walls (\(\Gamma _{\text {W,HG,iso}}\)).

The porous medium flow is characterized by the porosity \(\varphi \) of the material. We assume that the entire void space is connected. Thus, we do not have to deal with isolated cavities. Furthermore, the porosity is constant in the entire domain. For our two scenarios the flow in the porous medium is laminar. Furthermore, we have to account for the drag that evolves due to the solid obstacle using a Darcy-Forchheimer velocity equation. In short form, the resulting system of equations for a steady state flow readsIt consists of the continuity equation for the density \(\rho _f\) of the coolant, the Darcy-Forchheimer equation for the Darcy velocity \(\mathbf {v}_D\) and two heat equations for the temperatures \(T_s\) and \(T_f\) of the solid and the coolant, respectively, i.e.,Considering two temperatures allows to account for a potential thermal non-equilibrium of the solid and the fluid. The model is complemented with a perfect gas law for the coolant. Again, we refer to previous work  [4] for further details.

The model is supplemented with boundary conditions. Assuming that the conditions in the reservoir are constant and homogeneous we can derive the pressure and the temperatures on the reservoir boundary \(\Gamma _{\text {R}}\) from a fixed coolant temperature \(T_c\), e.g., given by measurements. Furthermore let \(T_b\) be the temperature of the solid on the backside of the porous material. Then the boundary conditions on the reservoir boundary \(\Gamma _{\text {R}}\) read \(T_s=T_b\), \(T_f= T_c\) and \(\rho _f =p_{\text {R}}/(R_c \, T_c)\), where \(R_c\) denotes the specific gas constant of the coolant. Furthermore, the solid side walls at \(\Gamma _{\text {W,PM}}\) are assumed to be adiabatic, i.e., \(\nabla T_s \cdot \mathbf {n} = \nabla T_f \cdot \mathbf {n} = 0\), and the slip conditions to hold, i.e., \(\mathbf {n} \cdot \mathbf {v}_D =0\). Note that at \(\Gamma _{\text {W,PM}}\) no boundary conditions for the density \(\rho _f\) have to be imposed because viscous effects are neglected in the continuity equation and the Darcy-Forchheimer equation.

So far we have not yet addressed boundary conditions at the coupling interface \(\Gamma _{\text {Int}}\) for the hot gas flow and the porous medium flow. Note that in the present work we only consider injection normal to the porous wall without entrainment of hot gas into the porous medium. At the coupling interface \(\Gamma _{\text {Int}}\) the flow quantities \(\mathbf {U}_{\text {HG}}\), see (2) or (3), in the hot gas domain \(\Omega _{\text {HG}}\) and \(\mathbf {U}_{\text {PM}}\), see (5), in the porous medium domain \(\Omega _{\text {PM}}\) are coupled. Since our model is based on a two-domain approach, we impose boundary conditions for both domains at the interface \(\Gamma _{\text {Int}}\) where the boundary conditions for the porous medium domain access data of the hot gas domain (PM \(\leftarrow \) HG) and vice versa (HG \(\leftarrow \) PM). The coupling conditions, which are specified in the following, make use of the equation of state and the formula for the mass flow rate, in general notation given by \(p=\rho R T\) and \(\dot{m}=\rho v A\), respectively.

PM \(\leftarrow \) HG: The coupling conditions at the interface \(\Gamma _{\text {Int}}\) for \(\Omega _{\text {PM}}\) use the pressure \(p_{\text {HG}}\), the temperature \(T_{\text {HG}}\) and the temperature gradient \(\nabla T_{\text {HG}}\) of the hot gas at the interface computed from the conservative quantities in \(\mathbf {U}_{\text {HG}}\). They are given bywhere \(v_{D,n}=\mathbf {v}_{D,n} \cdot \mathbf {n}\) denotes the absolute value of the injection velocity vector \(\mathbf {v}_{D,n}\) normal to the porous wall with outer unit normal vector \(\mathbf {n}\), \(\dot{m}_c\) the prescribed coolant mass flow rate and \(A_c\) the surface area of the porous medium, andHere \(\varvec{\kappa }_s\) and \(\kappa _{\text {HG}}\) denote the heat conduction tensor of the solid in the porous medium and the heat conduction coefficient of the hot gas fluid, respectively, and \(c_{p,f}\) the heat capacity at constant pressure of the hot gas fluid.

HG \(\leftarrow \) PM: The coupling conditions at the interface \(\Gamma _{\text {Int}}\) for \(\Omega _{\text {HG}}\) use the Darcy velocity normal component \(v_{D,n}\) and the fluid temperature \(T_f\) of the porous medium at the interface such thatIn contrast to original work where we investigated uniform mass injection  [3, 4, 8] we recently added the parameter \(\bar{o}\) to account for non-uniform injection that is set to 1 for uniform injection. Furthermore, the turbulence kinetic energy k at the interface is set to zero. The specific dissipation rate is determined byas proposed by Wilcox [29] to account for mass injection at the modeled porous surface. Here \(u_\tau = \sqrt{\tau _w / \rho }\) denotes the friction velocity with the wall shear stress \(\tau _w\).

For the supersonic nozzle flow the mass fractions of the coolant \(X_{\alpha ,c}\) are set at the interface to compute \(\rho _\alpha = \rho \cdot X_{\alpha ,c}\) with \(\rho \) from (8). Note that for foreign gas injection the mass fraction at the interface is 1 for the coolant species and 0 for all species contained in the hot gas flow.

We would like to emphasize that originally in [5] the injection velocity (6) was prescribed to simulate non-uniform injection. For a better control of a specific mass flow rate the parameter \(\bar{o}\) representing a local fraction or a percentage of a given mass flow was introduced in [16]. There it was incorporated in the coupling conditions for the porous medium. However, this caused modeling errors in the pressure distribution in the porous medium. Therefore, we recently incorporated this parameter in the coupling conditions for the hot gas [17].

Finally, we emphasize that in the case of uniform injection and thermal equilibrium in the porous material our coupling conditions enforce pressure continuity across the interface. For non-uniform injection and thermal equilibrium the local pressure continuity is lost because of the factor \(\bar{o}\) in (8), but due towe obtain integral pressure continuity instead. For thermal non-equilibrium, regardless of the injection type, the pressure is not continuous which is in agreement with the literature considering the interface as an idealization of a thin layer in which the pressure can change significantly because of the presence of the porous material  [23].

In the first scenario we investigate non-uniform injection coupling conditions and compare the results with uniform injection. This is motivated by developments in manufacturing allowing to design specific injection patterns of porous materials. This leads to the necessity of affordable non-uniform injection simulations for transpiration cooling. For this purpose, we consider the following configuration: the domains for the porous medium and the hot gas channel are \(\Omega _{\text {PM}}= 48\,\text {mm} \times 10.9\,\text {mm} \times 48\,\text {mm}\) and \(\Omega _{\text {HG}}= 1,120\,\text {mm} \times 60\,\text {mm} \times 90\,\text {mm}\), respectively, where the porous medium is mounted to the bottom side of the hot gas channel 580 mm downstream of the channel entrance and centrally in lateral direction. To investigate the influence of the injection pattern on the cooling and, thus, the effectivity of the transpiration cooling, we consider two injection patterns: the first one depends on the lateral coordinate z and is given by \(\bar{o} = 0.887 (\sin (\frac{5}{48} \pi \cdot (z+24))+1)\), while the second one depends on the streamwise coordinate x and is given by \(\bar{o} = 0.887 (\sin (\frac{5}{48} \pi \cdot x)+1)\). We refer to these as the lateral and the streamwise wave pattern, respectively. For \((x,z) \in [0,48] \times [-24,24]\) the wave patterns consist of three peaks and two sinks corresponding to high and low injection rates. The factor 0.887 scales \(\bar{o}\) such that (10) holds. A similar pattern was numerically and experimentally investigated in  [17] where \(\bar{o}\) depends on experimental outflow measurements. The disadvantage of \(\bar{o}\) depending on experimental data lies in the necessity to project the discrete data onto the computational mesh.

The porous medium domain is discretized by a mesh with \(60 \times 30 \times 60\) cells. The mesh lines are concentrated towards the hot gas and towards the reservoir side by applying a stretching technique.

For the hot gas domain we use a coarse mesh of about 135,000 cells in the first coupling step, i.e., Steps 2–5 of Algorithm 1. Then the data in the hot gas domain are prolongated to a uniformly refined mesh consisting of 1.08 million cells as an initial guess for the solution process on the finer mesh. Another four coupling steps are performed to obtain a converged coupled solution in the hot gas flow domain. Additionally a mesh convergence study was performed with a hot gas mesh of 8.64 million cells.

We perform three computations: uniform injection and non-uniform injection with lateral and streamwise wave pattern. Both the hot gas and the coolant are air. The parameters for the coupled simulations can be found in Table 1.

In Fig. 2 (left) we show the temperature distribution at the wall in the hot gas domain where the porous medium is mounted. In Fig. 2a, b the simulation results for the two non-uniform configurations are presented. We use the same integral mass flow rate for both the uniform and the non-uniform injection. Therefore, the porous areas with weak injection (light blue on the blowing insert in Fig. 2a, b are balanced by those with high injection. This has a strong effect on the cooling of the porous material itself and on the wake of the sample.

In Fig. 2a the areas with almost no injection can be seen over the porous material. In the wake of the sample the cooling film exhibits three streaks. The temperatures in the wake of areas with higher injection are lower than in the uniform injection case. In the wake of areas with almost no injection almost no cooling film is observed leading to high temperatures.

In Fig. 2b the areas with almost no injection can also be seen over the porous material. The cooling film in the wake of the sample is more uniform compared to the lateral wave pattern. The cooling film is slightly extended in streamwise direction compared with the uniform injection case.

To get a better understanding of the cooling film, especially in the wake of the sample, we perform a more detailed investigation by means of temperature data extracted in streamwise direction and lateral to the flow in the wake in Fig. 2 (right). A comparison of the wall temperature for the uniform injection and the lateral wave pattern is presented in Fig. 2d. Temperature data are extracted along the centerline (\(z =0\) mm) for the uniform injection and at a peak (\(z=0\) mm) and at a sink (\(z=14.4\) mm) for the lateral wave pattern. Due to almost no injection at the sink the temperature above the porous material is significantly higher than in the uniform case. It is also significantly higher in the wake (\(x>48\) mm), and at position \(x \approx 70\) mm the temperature increases to the isothermal wall temperature \(T_{\text {W}}=362.6\) K. For \(x>70\) mm the temperature is larger than \(T_{\text {W}}\) due to the adiabatic boundary condition in the wake of the porous medium and therefore no cooling film can be observed. For the peak the temperatures above the porous material are similar to the uniform injection. In the wake the temperatures are significantly lower due to a thicker cooling film resulting from the higher injection rate.

In Fig. 2e the uniform injection is compared with the streamwise wave pattern along the centerline (\(z=0\) mm). Above the porous material two areas with almost no injection can be identified for the streamwise wave pattern case. Here the temperatures are similar to the temperatures at the sink in the lateral wave pattern case. In the wake of the sample (\(x>48\) mm) the streamwise wave pattern shows a slightly thicker cooling film, probably due to the injection rate peak close to the trailing edge.

In Fig. 2f a comparison of uniform and non-uniform injection temperature data downstream of the sample (\(x=65\) mm) is presented. For the lateral wave pattern three troughs can be observed in the simulations. For the streamwise wave pattern the cooling film is uniform and very similar to the uniform injection. The mean values of the three curves are 348.4, 346.5 and 348.0 K for the lateral wave, streamwise wave and uniform case, respectively.

The second scenario is motivated by the nozzle test facility  [9, 21, 32] of the Shock Wave Laboratory at RWTH Aachen. The facility consists of a detonation tube, an attached axisymmetric nozzle, a vacuum tank and a damping section. The detonation tube provides hot water vapor for a quasi-steady nozzle flow for 7–10 ms. In this work, we concentrate on the simulation of the flow in the expansion part of the nozzle which has a length of 340 mm, a throat diameter of 16 mm, a half-opening angle of \(15^\circ \), a throat radius of curvature of 12 mm and an expansion ratio \(A_{\text {exit}}/A^*\) of 156. The nozzle has a slot with a length of 110.25 mm and a width of 8 mm in which a porous medium can be installed for transpiration cooling. The leading edge of the slot is positioned about 98 mm behind the nozzle throat (measured on the nozzle centerline).

We perform two simulations with (i) a porous medium with a cross-section surface corresponding to the dimensions of the slot and (ii) a porous medium of the same length but comprising the whole circumference of the nozzle. Due to the short test duration, analogously to the aforementioned isothermal nozzle wall there is no temperature increase of the fluid or the solid temperature in the porous material, i.e., we have \(T_f=T_s=T_c\). Hence, for our two simulations the coupling conditions (8) and (9) for the hot gas flow can be directly calculated without simulating the porous medium flow.

To account for the quasi-steady behavior, we couple the nozzle flow with the calculated porous medium data only once. In particular, we first of all perform a fully adaptive nozzle flow simulation without cooling which provides a final mesh with about 26 million cells. Based on the resulting hot gas data, (8) and (9) are calculated and used as boundary conditions for the subsequent slot and circumferential injection simulations, both with uniform cooling gas injection. These are then conducted by converging the flow solver once more on the final mesh.

The gas in the supersonic nozzle flow is water vapor with initial mass fractions \(X_{\text {H},\infty }=0.00251\), \(X_{\text {H}_2,\infty }=0.01176\), \(X_{\text {H}_2\text {O},\infty }=0.842\), \(X_{\text {O},\infty }=0.01231\), \(X_{\text {O}_2,\infty }=0.08668\) and \(X_{\text {OH},\infty }=0.04474\). The injected cooling gas is helium. The two simulations only differ in the dimensions of the porous medium and are conducted with the parameters listed in Table 2. Note that no parameters regarding the porous material are given because the porous medium flow is not simulated. The flow data are based on experiments planned at the Shock Wave Laboratory for a porous medium with dimensions \(110.25\,\text {mm} \times 8\,\text {mm} \times 7\,\text {mm}\) and characteristics similar to those of the first scenario, see Table 1.

The temperature distribution in the nozzle for slot and circumferential injection can be seen in Fig. 3a and b, respectively, on six slices through the nozzle. The particular positions of the slices are indicated in the figures. Note that only temperatures between 350 K and 1, 150 K are displayed. The leading and the trailing edge of the porous medium are positioned at \(x \approx 98\) mm and \(x \approx 208\) mm, respectively. The formation of a narrow cooling film in the case of slot injection is visible in Fig. 3a, in particular on the slices \(x=190\) mm and \(y=y_{\text {min}}+0.8\) mm, where \(y_{\text {min}}\) refers to the local y-coordinate on the nozzle wall at \(z=0\) depending on x. In contrast, in the case of circumferential injection, there is a continuous cooling film on the nozzle wall, see for instance the slices \(x=190\) mm or \(y=0\) mm in Fig. 3b. Since a cooling film is an obstacle to the hot gas flow, a stronger flow deceleration can be observed for the circumferential injection. Hence, the temperatures in the interior of the nozzle away from the walls are slightly higher compared with the slot injection case. This effect is visible in Fig. 3a and b, e.g., on slice \(y=0\) mm between \(x=190\) mm and \(x=280\) mm or on slice \(x=280\) mm.

An interesting phenomenon can be observed when comparing the temperatures in the cooling film boundary layers: With slot injection, lower temperatures can be achieved, as visible around the intersection point of the slices \(z=0\) mm, \(y=y_{\text {min}}+0.8\) mm and \(x=190\) mm. This is probably due to the slot being only a narrow obstacle which can be avoided by the hot gas flow by flowing around it. A comparison of the streamlines on the slice \(y=y_{\text {min}}+0.8\) mm in Fig. 3c, d confirms this conjecture. As a consequence, the slot cooling film can become thicker than the circumferential cooling film and, hence, results in a better cooling effectiveness at selective positions. Figure 3c, d also show the distribution of helium in the nozzle flow. In the slot injection case, the cooling film widens towards the trailing edge of the porous medium at \(x \approx 208\) mm and narrows downstream of the trailing edge.

Temperature boundary layer profiles in wall-normal direction in the wake of the porous medium at \(x=280\) mm on the centerline \(z=0\) mm are depicted in Fig. 4a. According to expectations, a better overall cooling is obtained by circumferential injection despite selective lower temperatures achieved by slot injection as described above. However, the comparison with the temperature profile for nozzle flow without any cooling gas injection reveals a significant cooling also in the slot injection case. The temperature peak in the lower part of the boundary layer (wall distance less than 2 mm) is reduced from 945 K (no cooling) to 853 K (slot injection) and 746 K (circumferential injection).

Figure 4b shows the wall heat fluxes for the line \(y=y_{\text {min}}\), \(z=0\) mm, which corresponds to the centerline of the slot, with and without cooling gas injection. Due to the isothermal nozzle walls, the wall temperature and the temperature of the cooling gas are the same (333.15 K). Hence, the wall temperature gradients up- and downstream of the leading edge of the porous medium are similar. However, the thermal conductivity of helium is about 3.5 times higher than that of water vapor which leads to the peak of the wall heat fluxes at the leading edge of the injection. Further downstream the heat fluxes rapidly decrease with a value close to zero at the trailing edge of the porous medium in the case of slot injection. The larger wall heat flux over the porous medium for circumferential injection is again explained by the thicker cooling film for slot injection. In the wake of the injection, both heat fluxes slowly increase, but the cooling effect persists up to the nozzle exit.