Extension of Analytical Wall Functions to Supersonic and Hypersonic Flows

Abstract

This research focuses on the development of wall functions suitable for the prediction of high-speed compressible flows, using RANS-based models. Wall-functions avoid the need for expensive fine near-wall meshes. The conventional log-law-based wall functions, however, have limitations even in incompressible cases, which are further compounded when applied to high-speed compressible flows. The analytical wall function proposed by Craft (Int. J. Heat Fluid Flow 23:148-160, 2002), which involves the analytical solution of simplified boundary-layer forms of the transport equations for the mean flow variables over the near-wall control volumes, has been successfully used in a range of incompressible flows. In this paper, a compressible flow version of the analytical wall function is proposed, which includes the following modifications: (a) improved variation of the convection terms in boundary-layer forms of the mean flow transport equations over the near-wall cells, (b) inclusion of thermal dissipation terms in the simplified analytical equation for the energy variation over the near-wall cells, (c) a more realistic representation of the thermal dissipation process in the discretized energy equation over the near-wall cells, (d) variable molecular viscosity (due to temperature variations) over the viscous sub-layer. The resultant model has been applied to shock wave/turbulent boundary layer interactions up to Mach numbers of 9 and comparisons are drawn with experimental data and with predictions from the log-law-based wall functions and from the Low-Re Launder and Sharma $k-\varepsilon$ model. The predictions resulting from the use of the compressible version of the analytical wall function are consistently closer to the data than those of other wall functions, and in some instances even better than those of corresponding low-Reynolds-number models. Improvements are especially noticeable in the prediction of the wall heat flux rates, where, in the highest Mach number cases, the log-law wall function generally predicts too low values in the shock interaction region, while the low-Reynolds-number models predict too high heat transfer rates, as a result of over-predicting turbulence levels in regions of extremely rapid near-wall temperature variations.

Appendix

1.1 a. Hydrodynamic Analytical Wall Function

In all subsequent equations, subscript “1” denotes the zero-turbulent viscosity sublayer ($y^*<y^*_v$), and subscript “2” the turbulent region of the turbulent layer ($y^*>y^*_v$).

The second integration of Eq. (18) with hyperbolic variation for the molecular viscosity over the zero turbulent viscosity sub-layer gives:

$$\begin{aligned} U_{1}=\, & {} \frac{1}{\mu _{v} } \left( \begin{array}{l} {\frac{D_{1} y^{*4} }{12} +\frac{C_{1} }{2} y^{*2} +A_{1} y^{*} } \\ {+b_{\mu } \left( \frac{D_{1} y^{*5} }{15} -\frac{D_{1} y_{v}^{*} }{12} y^{*4} +\frac{C_{1} }{3} y^{*3} +\frac{\left( A_{1} -C_{1} y_{v}^{*} \right) }{2} y^{*2} -A_{1} y_{v}^{*} y^{*} \right) } \end{array}\right) \end{aligned}$$

(30)

$$\begin{aligned} {U_2}=\, & {} \frac{1}{{{\mu _v}}}\left[ \begin{array}{l} \frac{{{D_2}}}{3}\frac{1}{{{\alpha ^3}}}\left[ {\frac{{{\alpha ^2}}}{3}{y^{ * 3}} - \frac{{\alpha \left( {1 - \alpha y_v^ * } \right) }}{2}{y^{ * 2}} + {{\left( {1 - \alpha y_v^ * } \right) }^2}{y^ * } - {{\left( {1 - \alpha y_v^ * } \right) }^3}\frac{1}{\alpha }\ln Y} \right] \\ + {C_2}\frac{1}{\alpha }\left[ {{y^ * } - \left( {1 - \alpha y_v^ * } \right) \frac{1}{\alpha }\ln Y} \right] + {A_2}\frac{1}{\alpha }\ln Y + {B_2} \end{array} \right] \end{aligned}$$

(31)

From the near wall cell boundary conditions, the coefficients in the above equations are:

$$\begin{aligned} {A_1}=\, & {} \frac{{{\mu _v}{U_n} - N}}{{\frac{{\ln {Y_n}}}{\alpha } + y_v^* - 0.5{b_\mu }y_v^{*2}}} \end{aligned}$$

(32)

$$\begin{aligned} {A_2}=\, & {} \left( {{D_1} - {D_2}} \right) \frac{{y_v^{ * 3}}}{3} + \left( {{C_1} - {C_2}} \right) y_v^ * + {A_1} \end{aligned}$$

(33)

$$\begin{aligned} {B_2}=\, & {} \frac{{{D_1}y_v^{*4}}}{{12}} + \frac{{{C_1}}}{2}y_v^{*2} + {A_1}y_v^* - {b_\mu }\left( {\frac{{{D_1}}}{{60}}y_v^{*5} + \frac{{{C_1}}}{6}y_v^{*3} + \frac{1}{2}{A_1}y_v^{*2}} \right) \nonumber \\{} & {} - \left[ {\frac{{{D_2}}}{3}\frac{1}{{{\alpha ^3}}}\left[ {\frac{{{\alpha ^2}}}{3}y_v^{*3} - \frac{{\alpha \left( {1 - \alpha y_v^ * } \right) }}{2}y_v^{*2} + {{\left( {1 - \alpha y_v^ * } \right) }^2}y_v^*} \right] + {C_2}\frac{1}{\alpha }y_v^*} \right] \end{aligned}$$

(34)

The coefficient N in (34) represents

$$\begin{aligned} N=\, & {} \frac{D_{2} }{3} \frac{1}{\alpha ^{3} } \left[ \frac{\alpha ^{2} }{3} y_{n}^{*3} -\frac{\alpha \left( 1-\alpha y_{v}^{*} \right) }{2} y_{n}^{*2} +\left( 1-\alpha y_{v}^{*} \right) ^{2} y_{n}^{*} -\left( 1-\alpha y_{v}^{*} \right) ^{3} \frac{1}{\alpha } \ln Y_{n} \right] \nonumber \\{} & {} {+C_{2} \frac{1}{\alpha } \left[ y_{n}^{*} -\left( \frac{1}{\alpha } -y_{v}^{*} \right) \ln Y_{n} \right] +\frac{1}{\alpha } \ln Y_{n} \left( \left( D_{1} -D_{2} \right) \frac{y_{v}^{*3} }{3} +\left( C_{1} -C_{2} \right) y_{v}^{*} \right) } \nonumber \\{} & {} +\left( \frac{D_{1} y_{v}^{*4} }{12} +\frac{C_{1} }{2} y_{v}^{*2} -b_{\mu } \left( \frac{D_{1} }{60} y_{v}^{*5} +\frac{C_{1} }{6} y_{v}^{*3} \right) \right) \nonumber \\{} & {} -\left[ \frac{D_{2} }{3} \frac{1}{\alpha ^{3} } \left[ \frac{\alpha ^{2} }{3} y_{v}^{*3} -\frac{\alpha \left( 1-\alpha y_{v}^{*} \right) }{2} y_{v}^{*2} +\left( 1-\alpha y_{v}^{*} \right) ^{2} y_{v}^{*} \right] +C_{2} \frac{1}{\alpha } y_{v}^{*} \right] \end{aligned}$$

(35)

where

$$\begin{aligned} {Y} =\, 1 + \alpha \left( {y^ * - y_v^ * } \right) \end{aligned}$$

(36)

1.2 b. Thermal Analytical Wall Function

In the zero-turbulent viscosity sublayer ($y^*<y^*_v$), the second integration of Eq. (22) with hyperbolic variation gives:

$$\begin{aligned} T_{1} =\frac{\Pr }{\mu _{v} } \left[ \begin{array}{l} {\frac{\textrm{D}_{th1} y^{*4} }{12} +A_{th1} y^{*} +b_{\mu } \left( \frac{\textrm{D}_{th1} y^{*5} }{15} -\frac{\textrm{D}_{th1} y_{v}^{*} }{12} y^{*4} +\frac{A_{th1} }{2} y^{*2} -A_{th1} y_{v}^{*} y^{*} \right) } \\ {-\int _{0}^{y^{*} }\frac{1}{C_{p} } \left( 1+b_{\mu } \left( y^{*} -y_{v}^{*} \right) \right) \mu U_{1} \frac{\partial U_{1} }{\partial y^{*} } dy^{*} } \end{array}\right] +T_{w} \end{aligned}$$

(37)

In the turbulent layer ($y^*>y^*_v$), the analytical temperature is:

$$\begin{aligned} T_{2} =\frac{\Pr }{\mu _{v} } \left\{ \begin{array}{l} {\frac{D_{th2} }{3} \frac{1}{\alpha _{t}^{3} } \left[ \frac{\alpha _{t}^{2} }{3} y^{*3} -\frac{\alpha _{t} \left( 1-\alpha _{t} y_{v}^{*} \right) }{2} y^{*2} +\left( 1-\alpha _{t} y_{v}^{*} \right) ^{2} y^{*} -\left( 1-\alpha _{t} y_{v}^{*} \right) ^{3} \frac{1}{\alpha _{t} } \ln Y_{T} \right] } \\ {+\frac{A_{th2} }{\alpha _{t} } \ln Y_{T} +B_{th2} -\frac{1}{\mu _{v} C_{p} } \int _{y_{v}^{*} }^{y^{*} }\frac{1}{Y_{T} } \mu _{v} U_{2} \left( \mu _{v} Y\frac{\partial U_{2} }{\partial y^{*} } \right) dy^{*} } \end{array}\right\} \end{aligned}$$

(38)

where the coefficients in the above equations are

$$\begin{aligned} {B_{th2}}= & {} {A_{th1}}y_v^* - \frac{1}{2}{b_\mu }{A_{th1}}y_v^{*2} + BT + \frac{{{\mu _v}}}{{\Pr }}{T_w} \end{aligned}$$

(39)

$$\begin{aligned} {A_{th2}}= & {} \left( {\frac{{{{\textrm{D}}_{th1}}}}{3} - \frac{{{D_{th2}}}}{3}} \right) y_v^{ * 3} + {A_{th1}} \end{aligned}$$

(40)

$$\begin{aligned} BT= & {} \frac{{{{\textrm{D}}_{th1}}y_v^{*4}}}{{12}} - \frac{{{{\textrm{D}}_{th1}}}}{{60}}{b_\mu }y_v^{*5} - \int _0^{y_v^*} {\frac{1}{{{C_p}}}\left( {1 + {b_\mu }\left( {{y^ * } - y_v^*} \right) } \right) \mu {U_1}\frac{{\partial {U_1}}}{{\partial {y^ * }}}d{y^ * }} \nonumber \\{} & {} - \frac{{{D_{th2}}}}{3}\frac{1}{{\alpha _t^3}}\left[ {\frac{{\alpha _t^2}}{3}y_v^{*3} - \frac{{{\alpha _t}\left( {1 - {\alpha _t}y_v^ * } \right) }}{2}y_v^{*2} + {{\left( {1 - {\alpha _t}y_v^ * } \right) }^2}y_v^*} \right] \end{aligned}$$

(41)

In order to complete the thermal wall function, the final expression for the wall temperature needs to be substituted into the code for prescribed heat-flux boundary conditions. From Eq. (39), when ${y^ * } = y_n^ *$, and $T_2=T_n$:

$$\begin{aligned} {T_w} = {T_n} - \frac{{\Pr }}{{{\mu _v}}}\left\{ \begin{array}{l} \frac{{{D_{th2}}}}{3}\frac{1}{{\alpha _t^3}}\left[ \begin{array}{l} \frac{{\alpha _t^2}}{3}y_n^{ * 3} - \frac{{{\alpha _t}\left( {1 - {\alpha _t}y_v^ * } \right) }}{2}y_n^{ * 2}\\ + {\left( {1 - {\alpha _t}y_v^ * } \right) ^2}y_n^ * - {\left( {1 - {\alpha _t}y_v^ * } \right) ^3}\frac{1}{{{\alpha _t}}}\ln {Y_{Tn}} \end{array} \right] \\ + \frac{{\ln {Y_{Tn}}}}{{{\alpha _t}}}\left( {\left( {\frac{{{{\textrm{D}}_{th1}}}}{3} - \frac{{{D_{th2}}}}{3}} \right) y_v^{ * 3} + {A_{th1}}} \right) + {A_{th1}}y_v^* - \frac{1}{2}{b_\mu }{A_{th1}}y_v^{*2} + BT\\ - \frac{1}{{{\mu _v}{C_p}}}\int _{y_v^ * }^{y_n^ * } {\frac{1}{{{Y_T}}}{\mu _v}{U_2}\left( {{\mu _v}Y\frac{{\partial {U_2}}}{{\partial {y^ * }}}} \right) d{y^ * }} \end{array} \right\} \end{aligned}$$

(42)

For the isothermal wall boundary conditions:

$$\begin{aligned} {q_{w}} = - \frac{{{\rho _v}{c_p}\sqrt{{k_P}} }}{{{\mu _v}}}{A_{th1}} \end{aligned}$$

(43)

where $A_{th1}$ can be obtained from (42) as:

$$\begin{aligned} {A_{th1}} = \frac{1}{{\left( {\frac{{\ln {Y_{Tn}}}}{{{\alpha _t}}} + y_v^* - \frac{{{b_\mu }}}{2}y_v^{*2}} \right) }}\left\{ \begin{array}{l} \frac{{{\mu _v}}}{{\Pr }}\left( {{T_n} - {T_w}} \right) \\ - \frac{{{D_{th2}}}}{3}\frac{1}{{\alpha _t^3}}\left[ \begin{array}{l} \frac{{\alpha _t^2}}{3}y_n^{ * 3} - \frac{{{\alpha _t}\left( {1 - {\alpha _t}y_v^ * } \right) }}{2}y_n^{ * 2}\\ + {\left( {1 - {\alpha _t}y_v^ * } \right) ^2}y_n^ * - {\left( {1 - {\alpha _t}y_v^ * } \right) ^3}\frac{1}{{{\alpha _t}}}\ln {Y_{Tn}} \end{array} \right] \\ - \frac{{\ln {Y_{Tn}}}}{{{\alpha _t}}}\left( {\frac{{{{\textrm{D}}_{th1}}}}{3} - \frac{{{D_{th2}}}}{3}} \right) y_v^{ * 3} - BT\\ + \frac{1}{{{\mu _v}{C_p}}}\int _{y_v^ * }^{y_n^ * } {\frac{1}{{{Y_T}}}{\mu _v}{U_2}\left( {{\mu _v}Y\frac{{\partial {U_2}}}{{\partial {y^ * }}}} \right) d{y^ * }} \end{array} \right\} \end{aligned}$$

(44)

where

$$\begin{aligned} {Y_{T}} = 1 + {\alpha _t}\left( {y^ * - y_v^ * } \right) = 1 + \alpha \frac{{\Pr }}{{{{\Pr }_t}}}\left( {y^ * - y_v^ * } \right) \end{aligned}$$

(45)