Prediction of aerodynamic flow fields using convolutional neural networks

Abstract

An approximation model based on convolutional neural networks (CNNs) is proposed for flow field predictions. The CNN is used to predict the velocity and pressure field in unseen flow conditions and geometries given the pixelated shape of the object. In particular, we consider Reynolds Averaged Navier–Stokes (RANS) flow solutions over airfoil shapes as training data. The CNN can automatically detect essential features with minimal human supervision and is shown to effectively estimate the velocity and pressure field orders of magnitude faster than the RANS solver, making it possible to study the impact of the airfoil shape and operating conditions on the aerodynamic forces and the flow field in near-real time. The use of specific convolution operations, parameter sharing, and gradient sharpening are shown to enhance the predictive capabilities of the CNN. We explore the network architecture and its effectiveness in predicting the flow field for different airfoil shapes, angles of attack, and Reynolds numbers.

Appendix: Governing equations

The RANS equations are derived by ensemble-averaging the conservation equations of mass, momentum and energy. These equations, for compressible flow are given by:

$$\begin{aligned}&\frac{\partial {{\bar{\rho }}}}{\partial t}+\frac{\partial \left( {{\bar{\rho }}}{{\hat{u}}}_i\right) }{\partial x_i} =0 \end{aligned}$$

(11)

$$\begin{aligned}&\frac{\partial \left( {{\bar{\rho }}}{{\hat{u}}}_i\right) }{\partial t}+\frac{\partial \left( {{\bar{\rho }}}{{\hat{u}}}_i{{\hat{u}}}_j\right) }{\partial x_j} =-\frac{\partial {{\bar{p}}}}{\partial x_i}+\frac{\partial {{\bar{\sigma }}}_{ij}}{\partial x_j}+\frac{\partial \tau _{ij}}{\partial x_j} \end{aligned}$$

(12)

$$\begin{aligned}&\frac{\partial \left( {{\bar{\rho }}}{{\hat{E}}}\right) }{\partial t}+\frac{\partial \left( {{\bar{\rho }}}{{\hat{H}}}{{\hat{u}}}_j\right) }{\partial x_j} = \frac{\partial }{\partial x_j}\left( {{\bar{\sigma }}}_{ij}{{\hat{u}}}_i+\overline{\sigma _{ij}u_i''}\right) \nonumber \\&\quad -\frac{\partial }{\partial x_j}\left( -{{\hat{\kappa }}}\frac{\partial {{\hat{T}}}}{\partial x_j}+c_P\overline{\rho u_j'' T''}-{{\hat{u}}}_i\tau _{ij}+\frac{1}{2}\overline{\rho u_i'' u_i'' u_j''}\right) , \end{aligned}$$

(13)

where the overbar indicates conventional time-average mean, $u_i$ is the fluid velocity, $\rho$ is the density, p is the pressure, $\tau _{ij}$ is the Reynolds stress term, $c_P$ is the heat capacity at constant pressure, and $\kappa$ is the kinetic energy of the fluctuating field (local turbulent kinetic energy). The density weighted time averaging (Favre averaging) of any quantity $\xi$, denoted by ${{\hat{\xi }}}$ is given as ${{\hat{\xi }}}=\overline{\rho \xi }/{{\bar{\rho }}}$, where,

$$\begin{aligned}&{{\hat{H}}}={{\hat{E}}}+\frac{{{\bar{p}}}}{{{\bar{\rho }}}}, \end{aligned}$$

(14)

$$\begin{aligned}&{{\bar{\sigma }}}_{ij}=\mu _t\left( \frac{\partial {{\hat{u}}}_i}{\partial x_j}+\frac{\partial {{\hat{u}}}_j}{\partial x_i}-\frac{2}{3}\frac{\partial {{\hat{u}}}_k}{\partial x_k}\delta _{ij}\right) , \end{aligned}$$

(15)

$$\begin{aligned}&\tau _{ij}=-\overline{\rho u_i'' u_j''}, \end{aligned}$$

(16)

$$\begin{aligned}&k=\frac{\widehat{u_i''^2}+\widehat{v_i''^2}+\widehat{w_i''^2}}{2}, \end{aligned}$$

(17)

$$\begin{aligned}&{{\bar{p}}} = (\gamma -1){{\bar{\rho }}}\left[ {{\hat{E}}} - \frac{{{\hat{u}}}^2+{{\hat{v}}}^2+{{\hat{w}}}^2}{2} - k\right] . \end{aligned}$$

(18)

To provide closure to the above equations, we use the model proposed by Spalart and Allmaras [39]. In this closure, the Boussinesq hypothesis relates the Reynolds stress and the effect of turbulence as an eddy viscosity $\mu _t$. Employing the Boussinesq approach, and Reynolds Analogy a transport equation for a working variable ${{\tilde{\nu }}}$ is solved to estimate the eddy viscosity field at every iteration.

$$\begin{aligned}&\frac{\partial {\tilde{\nu }}}{\partial t}+u_j\frac{\partial {\tilde{\nu }}}{\partial x_j} = C_{b1}\left[ 1-f_{t2}\right] {\tilde{S}}{\tilde{\nu }} \nonumber \\&\quad +~\frac{1}{\sigma }\left\{ \nabla \cdot \left[ \left( \nu +{\tilde{\nu }}\right) \nabla {\tilde{\nu }}\right] +C_{b2}\left| \nabla {\tilde{\nu }}\right| ^2\right\} \nonumber \\&\quad -\left[ C_{w1}f_w-\frac{C_{b1}}{\kappa ^2}f_{t2}\right] \left( \frac{{\tilde{\nu }}}{d}\right) ^2. \end{aligned}$$

(19)

The turbulent eddy viscosity is computed as $\mu _t={{\bar{\rho }}}{{\tilde{\nu }}} f_{v1}$, where,

$$\begin{aligned}&f_{v1}=\frac{\chi ^3}{\chi ^3+C_{v1}^3},~\chi =\frac{{\tilde{\nu }}}{\nu },~\nu =\frac{\mu }{{{\bar{\rho }}}},\\&f_{t2}=C_{t3} \exp \left( -C_{t4}\chi ^2\right) ,\\&{\tilde{S}}=S+\frac{{\tilde{\nu }}}{\kappa ^2 d^2}f_{v2},\\&S=\sqrt{2\varOmega _{ij}\varOmega {ij}},~f_{v2}=1-\frac{\chi }{1+\chi f_{v1}},\\&f_w=g\left[ \frac{1+C_{w3}^6}{g^6+C_{w3}^6}\right] ^{1/6}, \\&g=r+C_{w2}(r^6-r),~ r=\frac{{\tilde{\nu }}}{{\tilde{S}}\kappa ^2d^2}, \\&C_{w1}=\frac{C_{b1}}{\kappa ^2}+\frac{1+C_{b2}}{\sigma },~\\&C_{b1}=0.1355,~\sigma {=}2/3,~C_{b2} = 0.622,\\&\kappa =0.41,~C_{w2}=0.3,~ C_{w3}=2.0,~C_{v1}=7.1,~\\&C_{t3}=1.2,~C_{t4}=0.5. \end{aligned}$$

The first term on the right hand side of this Eq. 19 is the production term for ${{\tilde{\nu }}}$ while the second term represents dissipation.