3.1. Cartesian coordinates

The general solution of (2.7) in Cartesian coordinates can be obtained through a separation of variables and yields

(3.1)\begin{equation} \varPhi(x,y) = a + b y + ( \alpha \cos (\lambda x) + \beta \sin(\lambda x) ) ( \delta \cosh (\lambda y) + \gamma \sinh (\lambda y) ), \end{equation}

with the constants $a$, $b$, $\alpha$, $\beta$, $\delta$, $\gamma$ and $\lambda$ which have to be determined from specific boundary conditions. The porous domain, sketched in figure 1, starts at $x=y=0$ in the left bottom corner, and has a width of $d$ and a height of $t$. Boundaries 1 and 2 are vertical and represent the end of the porous domain in the horizontal direction. Boundary 3 represents the interface to an external flow field where an arbitrary pressure distribution $p_e(x)$ can be present. Boundary 4 is on the plenum side and can have any arbitrary thickness distribution $b_i(x)$. Any plenum pressure distribution $p_i(x)$ is possible, and this side can have a number of impermeable segments along this boundary (figure 1 only shows a single one for clarity). These boundary conditions can be represented mathematically as follows: the left and right boundary conditions are represented as impermeable conditions

(3.2)\begin{gather} \text{Boundary 1:} \ \frac{\partial \varPhi}{\partial x}(0,y)= 0, \end{gather}

(3.3)\begin{gather}\text{Boundary 2:} \ \frac{\partial \varPhi}{\partial x}(d,y)= 0, \end{gather}

which can also be interpreted as a symmetrical boundary. The interface to the external flow field has the pressure boundary condition

(3.4)\begin{equation} \text{Boundary 3:}\ \varPhi(x,t)= \varPhi_e(x) ={-} \frac{K_D}{\mu} p_e(x). \end{equation}

The plenum boundary is a mixed case of Dirichlet and Neumann boundary conditions. At permeable locations, the flow in the porous medium is driven by a pressure $p_i(x)$ which can vary along the horizontal axis. At impermeable locations, no flow can cross the boundary $b_i(x)$. This is mathematically described as

(3.5)\begin{equation} \text{Boundary 4:} \ \left. \begin{array}{ll@{}} \displaystyle \varPhi(x,b_i(x))=\varPhi_i(x) ={-} \dfrac{K_D}{\mu} p_i(x) & \mbox{for permeable locations,}\\ \displaystyle \varPsi(x,b_i(x)) = \varPsi_i = \mbox{const.} & \mbox{for impermeable locations.\ } \end{array}\right\}\end{equation}

Using (3.2) and (3.3) in (3.1) yields the intermediate solution

(3.6)\begin{equation} \varPhi(x,y) = a + b y + \sum_{n=1}^{\infty} ( \delta_n \cosh (N y) + \gamma_n \sinh (N y) ) \cos (N x) \quad \mbox{with\ } N = \frac{{\rm \pi} n}{d}. \end{equation}

An infinite sum of solutions has been constructed, where each individual term, i.e. for each $n$, fulfils (2.7). Due to the linearity of Laplace's equation, these can be superimposed, which is required in the next step when accounting for the internal and external boundary conditions. The external potential is expressed as the Fourier series

(3.7)\begin{align} \varPhi_e(x) &= \frac{A_0}{2} + \sum_{n=1}^{\infty} A_n \cos (Nx) \mbox{ with} \ A_0 = \frac{2}{d} \int_{0}^{d} \varPhi_e(x) \,\mbox{d}x \mbox{ and}\notag\\ A_n &= \frac{2}{d} \int_{0}^{d} \varPhi_e(x) \cos (Nx) \,\mbox{d}x .\end{align}

Using a term-by-term coefficient comparison between (3.6) and (3.7), each coefficient $A_n$ can be related to the respective coefficients $\delta _n$, $\gamma _n$, $a$ and $b$ by

(3.8)\begin{equation} \delta_n \cosh (N t) + \gamma_n \sinh (N t) = \frac{2}{d} \int_{0}^{d} \varPhi_e(x) \cos (Nx) \,\mbox{d}x, \end{equation}

and

(3.9)\begin{equation} a + b t = \frac{1}{d} \int_{0}^{d} \varPhi_e(x) \,\mbox{d}x,\end{equation}

yielding the solution

(3.10)\begin{align} \varPhi(x,y) & = \frac{1}{d} \int_{0}^{d} \varPhi_e(x)\,\mbox{d}x + B_0(y-t) \nonumber\\ & \quad + \sum_{n=1}^{\infty} B_n ( \cosh (N y) -\coth (N t) \sinh (N y) ) \cos (N x) \nonumber\\ & \quad + \sum_{n=1}^{\infty} \frac{\displaystyle\frac{2}{d} \int_{0}^{d} \varPhi_e(x) \cos (Nx) \,\mbox{d}x}{\sinh (N t)} \sinh (N y) \cos (N x). \end{align}

For convenience, in the following derivation, the constants $b$ and $\delta _n$ have been relabelled as $B_0$ and $B_n$, respectively. The corresponding streamfunction can be obtained via the Cauchy–Riemann equations ((2.8a,b)) as

(3.11)\begin{align} \varPsi(x,y) & = B_{{-}1} - B_0 x - \sum_{n=1}^{\infty} B_n ( \sinh (N y) -\coth (N t) \cosh (N y) ) \sin(N x) \nonumber\\ &\quad - \sum_{n=1}^{\infty} \frac{\displaystyle\frac{2}{d} \int_{0}^{d} \varPhi_e(x) \cos (Nx) \,\mbox{d}x}{\sinh (N t)} \cosh (N y) \sin(N x), \end{align}

with the arbitrary constant $B_{-1}$ which stems from an indefinite integral. The remaining unknown coefficients $B_n$ are determined from the internal boundary condition by employing and expanding on the methodology developed in Read (Reference Read2007). The boundary forcing function $F(x)$ is defined as

(3.12) \begin{equation} F(x) = \left\{ \begin{array}{@{}ll} \displaystyle\varPhi_i(x) - \dfrac{1}{d} \int_{0}^{d} \varPhi_e(x)\,\mbox{d}x \\ \quad - \displaystyle \sum_{n=1}^{\infty} \dfrac{\displaystyle\dfrac{2}{d} \int_{0}^{d} \varPhi_e(x) \cos (Nx) \,\mbox{d}x}{\sinh (N t)} \sinh (N b_i(x)) \cos (N x), & \mbox{perm.}\\ \displaystyle\varPsi_i + \sum_{n=1}^{\infty} \dfrac{\displaystyle\dfrac{2}{d} \int_{0}^{d} \varPhi_e(x) \cos (Nx) \,\mbox{d}x}{\sinh (N t)} \cosh (N b_i(x)) \sin(N x). & \mbox{imperm}. \end{array} \right. \end{equation}

In addition, the coefficient function $U_n(x)$ is defined as

(3.13)\begin{equation} U_{n}(x) = \left\{ \begin{array}{@{}ll} ( \cosh (N b_i(x)) -\coth (N t) \sinh (N b_i(x)) ) \cos (N x) , & \mbox{permeable} \\ - ( \sinh (N b_i(x)) -\coth (N t) \cosh (N b_i(x)) ) \sin(N x), & \mbox{impermeable}, \end{array} \right.\end{equation}

with

(3.14)\begin{equation} U_{0}(x) = \left\{ \begin{array}{@{}ll} b_i(x) - t , & \mbox{permeable} \\ - x, & \mbox{impermeable}, \end{array} \right. \end{equation}

and

(3.15)\begin{equation} U_{{-}1}(x) = \left\{ \begin{array}{@{}ll} 0, & \mbox{permeable} \\ 1, & \mbox{impermeable}. \end{array} \right. \end{equation}

These functions allow the expression of (3.5) as

(3.16)\begin{equation} F(x) = \sum_{n={-}1}^{M} B_n U_n(x).\end{equation}

Here, the Fourier series is truncated to $M+2$ terms corresponding to the coefficients $B_{n}$ for $n = -1,0,1,\ldots ,M$. Following on from the work in Read (Reference Read2007), the normal equations of this problem are

(3.17)\begin{equation} \sum_{j={-}1}^{M} B_j \int_{0}^{d} U_i(x) U_j(x)\, \mathrm{d} x = \int_{0}^{d} U_i(x) F(x)\, \mathrm{d} x,\end{equation}

which can also be represented in a matrix form as

(3.18)\begin{equation} \boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{b} = \boldsymbol{f},\end{equation}

with the matrix entries for $\boldsymbol {U}$,

(3.19)\begin{equation} [\boldsymbol{U}]_{ij} = \int_{0}^{d} U_i(x) U_j(x)\, \mathrm{d} x ,\end{equation}

the vector entries for $\boldsymbol {f}$,

(3.20)\begin{equation} [\boldsymbol{f}]_{i} = \int_{0}^{d} U_i(x) F(x)\, \mathrm{d} x ,\end{equation}

and the entries of the unknown coefficients’ vector $\boldsymbol {b}$,

(3.21)\begin{equation} [\boldsymbol{b}]_{i} = B_i .\end{equation}

Equations (3.19) and (3.20) can be solved with any appropriate numerical method. In the cases presented in § 5, the integrals have been solved using trapezoidal integration. Equation (3.18) can simply be solved by

(3.22)\begin{equation} \boldsymbol{b} = \boldsymbol{U}^{{-}1} \boldsymbol{\cdot} \boldsymbol{f},\end{equation}

to yield the remaining coefficients $B$. Equation (3.22) is valid for a square matrix of size $M+2 \times M+2$. Alternatively, the matrix can be rectangular and of the size $M+2 \times \tilde {M}$, which results in the solution

(3.23)\begin{equation} \boldsymbol{U}^\textrm{T} \boldsymbol{U} \boldsymbol{b} = \boldsymbol{U}^\textrm{T} \boldsymbol{\cdot} \boldsymbol{f}. \end{equation}

This over collocation method has been shown to work best with $\tilde {M} = 2M$ to $5M$ points (Read Reference Read2007). For convenience, the resulting formulations for the pressure field $p(x,y)$, velocity components $v_x(x,y)$, $v_y(x,y)$ and the streamfunction $\varPsi (x,y)$ are given below in terms of the original pressure boundary conditions. Pressure field

(3.24)\begin{align} p(x,y) & = \frac{1}{d} \int_{0}^{d} p_e(x)\,\mbox{d}x - \frac{\mu}{K_D} B_0(y-t) \nonumber\\ &\quad - \frac{\mu}{K_D} \sum_{n=1}^{M} B_n ( \cosh (N y) -\coth (N t) \sinh (N y) ) \cos (N x) \nonumber\\ & \quad + \sum_{n=1}^{M} \frac{\displaystyle\frac{2}{d} \int_{0}^{d} p_e(x) \cos (Nx) \,\mbox{d}x}{\sinh (N t)} \sinh (N y) \cos (N x) , \end{align}

velocity components

(3.25)\begin{align} v_x(x,y) &={-} \sum_{n=1}^{M} B_n N ( \cosh (N y) -\coth (N t) \sinh (N y) ) \sin(N x) \nonumber\\ & \quad + \frac{K_D}{\mu} \sum_{n=1}^{M} N \frac{\displaystyle\frac{2}{d} \int_{0}^{d} p_e(x) \cos (Nx) \,\mbox{d}x}{\sinh (N t)} \sinh (N y) \sin(N x) ,\end{align}

(3.26)\begin{align} v_y(x,y) & = B_0 + \sum_{n=1}^{M} B_n N ( \sinh (N y) -\coth (N t) \cosh (N y) ) \cos (N x)\notag\\ &\quad - \frac{K_D}{\mu} \sum_{n=1}^{M} N \frac{\displaystyle\frac{2}{d} \int_{0}^{d} p_e(x) \cos (Nx) \,\mbox{d}x}{\sinh (N t)} \cosh (N y) \cos (N x) , \end{align}

and streamfunction

(3.27)\begin{align} \varPsi(x,y) & = B_{{-}1} - B_0 x - \sum_{n=1}^{M} B_n ( \sinh (N y) -\coth (N t) \cosh (N y) ) \sin(N x) \nonumber\\ & \quad + \frac{K_D}{\mu} \sum_{n=1}^{M} \frac{\displaystyle\frac{2}{d} \int_{0}^{d} p_e(x) \cos (Nx) \,\mbox{d}x}{\sinh (N t)} \cosh (N y) \sin(N x), \end{align}

with $N = ({\rm \pi} n )/d$.

Figure 1.

Sketch of mathematical domain in Cartesian coordinates. White areas denote porous material, while shaded areas denote impermeable material.

3.2. Cylindrical coordinates

The derivation of the solution in cylindrical coordinates is analogous to the previous section. Therefore, only the main steps will be shown. The general solution of (2.7) in cylindrical coordinates is

(3.28)\begin{equation} \varPhi(\theta,r) = a + b\ln (r) + ( \alpha \cos (\lambda \theta) + \beta \sin(\lambda \theta) ) ( \delta r^{-\lambda} + \gamma r^{\lambda} ). \end{equation}

The respective porous domain is shown in figure 2 and resembles an architecture which would be found in a porous leading edge of a hypersonic vehicle (Colwell & Modlin Reference Colwell and Modlin1992). The porous domain extends from $\theta =0$ to $s$, where it is terminated by an impermeable boundary in radial orientation, and the domain is symmetrical along the $\theta =0$ axis. Similarly to the Cartesian case, the external boundary is given by a pressure distribution $p_e(\theta )$ at the external radius $R_e$. The internal boundary can have an arbitrary shape $R_i(\theta )$, and can feature permeable and impermeable sections. Theses boundary conditions can be expressed as

(3.29)\begin{gather} \text{Boundary 1:} \ \frac{\partial \varPhi}{\partial \theta}(0,r)= 0, \end{gather}

(3.30)\begin{gather}\text{Boundary 2:} \ \frac{\partial \varPhi}{\partial \theta}(s,r)= 0, \end{gather}

(3.31)\begin{gather}\text{Boundary 3:}\ \varPhi(\theta,R_e)= \varPhi_e(\theta) ={-} \frac{K_D}{\mu} p_e(\theta), \end{gather}

(3.32)\begin{gather}\text{Boundary 4:} \ \left. \begin{array}{ll@{}} \displaystyle \varPhi(\theta,R_i(\theta))=\varPhi_i(\theta) ={-} \dfrac{K_D}{\mu} p_i(\theta) & \mbox{for permeable locations,}\\ \displaystyle \varPsi(\theta,R_i(\theta)) = \varPsi_i = \mbox{const.} & \mbox{for impermeable locations.} \end{array}\right\} \end{gather}

Implementing boundary conditions 1–3 yields the potential function

(3.33)\begin{align} \varPhi(\theta,r) & = \frac{1}{s} \int_{0}^{s} \varPhi_e(\theta)\,\mbox{d}\theta + B_0 \ln \left(\frac{r}{R_e}\right) + \sum_{n=1}^{\infty} B_n ( r^{{-}N} -R_e^{{-}2N} r^N ) \cos (N \theta) \nonumber\\ & \quad +\sum_{n=1}^{\infty} \left(\frac{r}{R_e}\right)^N \cos(N \theta) \frac{2}{s} \int_{0}^{s} \varPhi_e(\theta) \cos (N\theta) \,\mbox{d}\theta , \end{align}

and the streamfunction

(3.34)\begin{align} \varPsi(\theta,r) & = B_{{-}1} - B_0 \theta + \sum_{n=1}^{\infty} B_n ( r^{{-}N} + R_e^{{-}2N} r^N ) \sin(N \theta) \nonumber\\ &\quad - \sum_{n=1}^{\infty} \left(\frac{r}{R_e}\right)^N \sin(N \theta) \frac{2}{s}\int_{0}^{s} \varPhi_e(\theta) \cos (N\theta) \,\mbox{d}\theta. \end{align}

In order to account for boundary condition 4, the following functions are defined. The forcing function

(3.35) \begin{equation} F(x) = \left\{ \begin{array}{@{}ll} \displaystyle\varPhi_i(\theta) - \dfrac{1}{s} \int_{0}^{s} \varPhi_e(\theta)\,\mbox{d}\theta \\ \quad - \displaystyle \sum_{n=1}^{\infty} \left(\dfrac{R_i(\theta)}{R_e}\right)^N \cos(N \theta)\dfrac{2}{s} \int_{0}^{s} \varPhi_e(\theta) \cos (N\theta) \,\mbox{d}\theta, & \mbox{perm.} \\ \displaystyle\varPsi_i + \sum_{n=1}^{\infty} \left(\dfrac{R_i(\theta)}{R_e}\right)^N \sin(N \theta) \dfrac{2}{s} \int_{0}^{s} \varPhi_e(\theta) \cos (N\theta) \,\mbox{d}\theta, & \mbox{imperm}. \end{array} \right.\end{equation}

and the coefficient function

(3.36)\begin{equation} U_{n}(\theta) = \left\{ \begin{array}{@{}ll} ( R_i(\theta)^{{-}N} -R_e^{{-}2N} R_i(\theta)^N ) \cos (N \theta), & \mbox{permeable}, \\ ( R_i(\theta)^{{-}N} +R_e^{{-}2N} R_i(\theta)^N ) \sin(N \theta), & \mbox{impermeable}, \end{array} \right.\end{equation}

with

(3.37)\begin{equation} U_{0}(\theta) = \left\{ \begin{array}{@{}ll} \ln \left(\dfrac{R_i(\theta)}{R_e}\right) , & \mbox{permeable} \\ - \theta, & \mbox{impermeable}, \end{array} \right.\end{equation}

and

(3.38)\begin{equation} U_{{-}1}(\theta) = \left\{ \begin{array}{@{}ll} 0, & \mbox{permeable}, \\ 1, & \mbox{impermeable}. \end{array} \right.\end{equation}

With these functions defined, the steps from (3.16) to (3.22) have to be carried out to obtain the unknown coefficients $B_n$. The resulting formulations for the pressure field $p(\theta ,r)$, velocity components $v_{\theta }(\theta ,r)$, $v_r(\theta ,r)$ and the streamfunction $\varPsi (\theta ,r)$ are given below in terms of the original pressure boundary conditions. Pressure field

(3.39)\begin{align} p(\theta,r) & = \frac{1}{s} \int_{0}^{s} p_e(\theta)\,\mbox{d}\theta - \frac{\mu}{K_D} B_0 \ln \left(\frac{r}{R_e}\right) - \frac{\mu}{K_D} \sum_{n=1}^{M} B_n ( r^{{-}N} -R_e^{{-}2N} r^N ) \cos (N \theta)\nonumber\\ & \quad + \sum_{n=1}^{M} \left(\frac{r}{R_e}\right)^N \cos(N \theta)\frac{2}{s} \int_{0}^{s} p_e(\theta) \cos (N\theta) \,\mbox{d}\theta , \end{align}

velocity components

(3.40)\begin{align} v_{\theta}(\theta,r) &={-} \sum_{n=1}^{M} B_n N ( r^{{-}N-1} -R_e^{{-}2N} r^{N-1} ) \sin(N \theta) \nonumber\\ & \quad +\frac{K_D}{\mu} \sum_{n=1}^{M} \frac{r^{N-1}}{R_e^N} N \sin (N \theta) \frac{2}{s} \int_{0}^{s} p_e(\theta) \cos (N\theta) \,\mbox{d}\theta , \end{align}

(3.41)\begin{align} v_r(\theta,r) & = B_0 \frac{1}{r} - \sum_{n=1}^{M} B_n N ( r^{{-}N-1} +R_e^{{-}2N} r^{N-1} ) \cos (N \theta) \nonumber\\ &\quad -\frac{K_D}{\mu} \sum_{n=1}^{M} \frac{r^{N-1}}{R_e^N} N \cos (N \theta) \frac{2}{s} \int_{0}^{s} p_e(\theta) \cos (N\theta) \,\mbox{d}\theta , \end{align}

and streamfunction

(3.42)\begin{align} \varPsi(\theta,r) & = B_{{-}1} - B_0 \theta + \sum_{n=1}^{M} B_n ( r^{{-}N} + R_e^{{-}2N} r^N ) \sin(N \theta) \nonumber\\ & \quad +\frac{K_D}{\mu} \sum_{n=1}^{M} \left(\frac{r}{R_e}\right)^N \sin(N \theta) \frac{2}{s}\int_{0}^{s} \varPhi_e(\theta) \cos (N\theta) \,\mbox{d}\theta, \end{align}

with $N = ({\rm \pi} n )/s$.

Figure 2.

Sketch of the mathematical domain in cylindrical coordinates. White areas denote porous material, shaded areas denote impermeable material and dashed lines denote symmetry.

5.1. Porous flow in flat plate domain

Figure 3 shows a sketch of the porous domain. A porous flat plate of width $d=12$ mm and thickness $t=2$ mm is bounded left and right by impermeable materials. The domain features a linearly decreasing wall thickness from $b_i(x=6\ \mbox {mm})=0$ mm to $b_i(x=12\ \mbox {mm})=0.67$ mm. The plenum side is separated into two chambers where a constant pressure of $4$ bar is applied from $x=0$ mm to $3$ mm, and a constant pressure of $1.5$ bar is applied from $x=6$ mm to $12$ mm. The space between these domains is occupied by an impermeable boundary. The top side of the plate is subjected to a pressure boundary that decreases nonlinearly from $1.5$ bar to $0.75$ bar over the length of the plate and is described by

(5.1)\begin{equation} p_e(x) = 1.5\cdot 10^5 \frac{d}{x + d} \ \mathrm{Pa}.\end{equation}

The porous material features a permeability of $K_D=2.5\cdot 10^{-14}$ m$^2$, and the fluid features a dynamic viscosity of $\mu =1.5\cdot 10^{-5}\ \textrm {Pa}\cdot \textrm {s}$. The resulting pressure and velocity fields are shown in figure 4 which have been determined by evaluating (3.24)–(3.26), where the absolute velocity is determined as

(5.2)\begin{equation} v(x,y) = \sqrt{v_x(x,y)^2 + v_y(x,y)^2}.\end{equation}

The value of the computed Fourier coefficients $M$ in this case was 80. A comprehensive study of the error in each term for different values of $M$ can be found in Read (Reference Read2007). The high pressure boundary between $x=0$ and $3$ mm on the plenum side is the main driving feature in the pressure distribution. The behaviour between $x=0$ and $2$ mm approximates a one-dimensional flow in the $y$-direction as the change in external pressure over this region is small. Significant edge effects are visible in the vicinity of the pressure-impermeability boundary at $x=3$ mm. The velocity distribution clearly shows this boundary between different domains as peaks in the field. Both the left ($x=3$ mm) and the right ($x=6$ mm) boundaries are clearly visible where a large velocity concentration is induced by a strong pressure gradient in the flow field.

Figure 3.

Sketch of flat plate example (a) with pressure boundary conditions (b).

Figure 4.

Pressure and velocity field in porous domain of flat plate case.

The domain has been numerically simulated with the commercial multiphysics tool COMSOL using a mesh of approximately $5000$ cells. A mesh independence study was carried out which showed that a finer mesh leads to a maximum difference of 0.5 % in the velocity field when the mesh size is doubled. This is deemed as an acceptable accuracy and the results discussed in the following relate to the coarser mesh. Figure 5 shows the relevant values obtained at the internal and external boundaries. As per boundary condition 4, the streamfunction is constant over the range of $3$ to $6$ mm, which represents a zero velocity case due to the impermeable backside. The numerically and analytically obtained pressure distributions show an excellent agreement. At sharp edges, e.g. $x=3$ mm, the Gibbs phenomenon of the Fourier series is visible, leading to slight oscillations in the vicinity of these discontinuities. Strategies to minimise this phenomenon are discussed in Read (Reference Read2007), with the best approach being over collocation or discrete least squares when solving (3.23). A similar level of agreement is also present for the external outflow velocity. The velocity distribution is increasing from $x=0$ mm to a peak at approximately $x=1.5$ mm. This peak is a result of the falling external pressure with larger $x$ (see figure 3) while the driving pressure distribution inside the porous material stays almost constant. Beyond the peak, the diffusion around the edge of the plenum ($p_1$) leads to a drop in the external outflow velocity, as the low pressure region above the impermeable boundary ($x=3$ to $6$ mm) redirects a significant amount of the coolant flow into a lateral direction. Toward the rear of the plate ($x>6$ mm) the outflow velocity starts to increase again as the second plenum ($p_2$) leads to a coolant flow through the porous domain. The decreasing external pressure boundary for larger $x$, and the thinner wall lead to a larger outflow velocity, as the driving plenum pressure remains constant.

Figure 5.

Flow field along boundaries $y=b_i(x)$ and $y=t$. (a) Pressure and streamfunction at internal boundary $y = b_i(x)$ and (b) velocity magnitude at external boundary $y=t$.

5.2. Porous flow in cylindrical leading edge

Figure 6 shows a sketch of the cylindrical test case investigated in this work. The geometry represents a possible porous leading edge architecture and is symmetrical around $\theta =0$. The angular extend ($s$) is a quarter circle with a constant external radius of $R_e=6$ mm. The porous domain ends at $\theta ={\rm \pi} /2$ where an impermeable boundary exists. The internal radius is $4.5$ mm except from $0.3 \cdot {\rm \pi}/2$ to the symmetry plane ($\theta =0$) where the wall thickness gradually decreases from $R_i(\theta =0.3 \cdot {\rm \pi}/2)= 4.5$ mm in a straight line to a minimum of $R_i(\theta =0)= 5.4$ mm. The internal boundary is split into two plenum chambers with constant pressures of $5$ bar and $3$ bar, respectively. Between $\theta =0.3\cdot {\rm \pi}/2$ and $0.75\cdot {\rm \pi}/2$, an impermeable boundary prevents coolant from entering into the porous material. The external pressure decreases from $3.65$ bar at the symmetry plane to $0.05$ bar at the end of the porous boundary and is defined by

(5.3)\begin{equation} p_e(\theta) = 5 \cdot 10^3\ \mbox{Pa} + 3.6 \cdot 10^5 \cdot \cos^2(\theta) \ \mbox{Pa} . \end{equation}

A similar pressure distribution would be expected from a post-shock flow field around a leading edge wedge geometry. The porous material features a permeability of $K_D=2.5\cdot 10^{-14}$ m$^2$, and the fluid features a dynamic viscosity of $\mu =1.5\cdot 10^{-5}\ \textrm {Pa} \cdot \textrm {s}$. The value of computed Fourier coefficients $M$ in this case was 100. The resulting pressure and velocity fields are presented in figure 7 where the velocity shown is the absolute value determined through

(5.4)\begin{equation} v(\theta,r) = \sqrt{v_\theta(\theta,r)^2 + v_r(\theta,r)^2}.\end{equation}

Similar to the Cartesian flat plate case, the pressure distribution leads to large gradients between the permeable and impermeable boundaries at the internal plenum side. These then result in a strong concentration of velocity at these boundaries. In figure 8, the internal pressure and streamfunction distributions and the external outflow velocity are presented. The constant streamfunction at the internal boundary between the arc lengths of $2.5$ and $5.5$ mm confirms the impermeable boundary, and the pressure distribution outside of this region also correctly replicates the required constant plenum pressures. Figure 8 also contains a numerical comparison, that relates to the result of a COMSOL simulation using a mesh of approximately $10\,000$ cells. A mesh independence study was carried out which showed that a finer mesh leads to a maximum difference of 0.2 % in the velocity field when the mesh size is doubled. This is deemed as an acceptable accuracy and the results discussed in the following relate to the coarser mesh. The agreement of the pressure and velocity values at the shown boundaries is excellent with the exception of slight oscillations at an arc length of $0$ mm due to the Gibbs phenomenon. In this case, the steep decrease of the outflow velocity away from the symmetry plane is mainly due to the rapidly increasing thickness of the porous domain. The rival effect of a lower pressure for greater $\theta$ cannot compete with the influence of a thinner wall. Naturally, the outflow velocity is lowest above the impermeable section, as this region redirects a majority of the fluid flow laterally, which would have otherwise contributed to the radial outflow. Toward the rear of the domain at arc lengths $>5$ mm, the outflow velocity increases again due to the much lower external pressure (see figure 6).

Figure 6.

Sketch of cylindrical example (a) with pressure boundary conditions (b).

Figure 7.

Pressure and velocity field in porous domain of wedge leading edge case.

Figure 8.

Flow field along boundaries $r=R_i(\theta )$, and $r=R_e$. (a) Pressure and streamfunction at internal boundary $r=R_i(\theta )$ and (b) velocity magnitude at external boundary $r=R_e$.