Practical implementation aspects of Galerkin reduced order models based on proper orthogonal decomposition for computational fluid dynamics

Abstract

The significant computational cost of computational fluid dynamics (CFD) is still the major drawback to its use for multi-query analysis or optimization cycles. This paper is intended to present the formulation of a Reduced Order Model (ROM) for CFD through the traditional Galerkin projection of differential equations on the modes obtained by proper orthogonal decomposition (POD). The ROM framework was built considering the projection of the continuous equations, which allows its utilization based solely on the Full Order Model (FOM) results. This strategy enables the independence of the CFD solver. This preliminary research illustrates the application of this technique in three examples of increasing complexity. Firstly, the linear heat equation in a one-dimensional domain, then the one-dimensional Burgers’ equation. Finally, the Navier–Stokes equations in a two-dimensional domain for a backward-facing step flow is analyzed considering different Reynolds numbers. The implementation details are discussed and, using ANSYS CFX^® as FOM reference, the results for the incompressible Navier–Stokes are investigated in terms of the accuracy level achieved, depending on the parameters chosen for the snapshots generation and the execution time reduction obtained.

Appendix A: expansion of Galerkin projection of the Navier–Stokes equations

Considering a domain Ω and the tensors obtained from the Galerkin projections of the Navier–Stokes equations in Sect. 3.1:

$$A_{ij} = - \left\langle {{\varvec{\uppsi}}_{j} \cdot \nabla \overline{{\mathbf{U}}} \,,\,{\varvec{\uppsi}}_{i} } \right\rangle - \left\langle {\overline{{\mathbf{U}}} \cdot \nabla {\varvec{\uppsi}}_{j} \,,\,{\varvec{\uppsi}}_{i} } \right\rangle - \frac{1}{Re}\left\langle {\nabla {\varvec{\uppsi}}_{j} ,\nabla {\varvec{\uppsi}}_{i} } \right\rangle , \, \, i, \, j \, = \, 1, \ldots ,m$$

(A.1)

$$N_{ijk} = - \left\langle {{\varvec{\uppsi}}_{j} \cdot \nabla {\varvec{\uppsi}}_{k} \,,\,{\varvec{\uppsi}}_{i} } \right\rangle , \, i, \, j, \, k \, = \, 1, \ldots ,m$$

(A.2)

$$e_{i} = - \left\langle {\overline{{\mathbf{U}}} \cdot \nabla \overline{{\mathbf{U}}} \,,\,{\varvec{\uppsi}}_{i} } \right\rangle - \frac{1}{Re}\left\langle {\nabla \overline{{\mathbf{U}}} ,\nabla {\varvec{\uppsi}}_{i} } \right\rangle , \, i \, = \, 1, \ldots ,m$$

(A.3)

$$\alpha_{0i} = \left\langle {{\mathbf{U}}_{0} - \overline{{\mathbf{U}}} ,{\varvec{\uppsi}}_{i} } \right\rangle , \, i \, = \, 1, \ldots ,m\,$$

(A.4)

These tensors have to be expanded in the context of two-dimensional velocity fields to be computationally implemented. For the expansion, it is worth recalling:

$${\mathbf{U}} = \,\left( {\,U\,\,\,\,V\,} \right)\,;\;\overline{{\mathbf{U}}} = \,\left( {\,\overline{U} \,\,\,\,\overline{V\,} \,} \right);\;{\mathbf{U}}_{0} = \left( {\,U_{0} \,\,\,\,V_{0} \,} \right)\; {\text \, \rm {and}} \, \;{\varvec{\uppsi}} = \left( {\,\psi_{U} \,\,\,\psi_{V} } \right);$$

(A.5)

and the definition of inner product for two-dimensional functions:

$$\left\langle {{\mathbf{f}},{\mathbf{g}}} \right\rangle = \int\limits_{\varOmega } {\left( {f_{x} g_{x} + f_{y} g_{y} } \right)\,\,dxdy}$$

(A.6)

Firstly, expanding the first order tensors, by starting with the vector which corresponds to the initial condition:

$$\alpha_{0i} = \left\langle {{\mathbf{U}}_{0} - \overline{{\mathbf{U}}} ,{\varvec{\uppsi}}_{i} } \right\rangle = \int\limits_{\varOmega } {\left[ {\left( {U_{0} - \overline{U} } \right)\psi_{Ui}^{{}} + \left( {V_{0} - \overline{V} } \right)\psi_{Vi}^{{}} } \right]\,} dxdy$$

(A.7)

The vector e will be decomposed as $e_{i} = - e_{1i} - \tfrac{1}{Re}e_{2i}$ and the resulting expansions will be:

$$\begin{gathered} e_{1i} = \left\langle {\overline{{\mathbf{U}}} \cdot \nabla \overline{{\mathbf{U}}} \,,\,{\varvec{\uppsi}}_{i} } \right\rangle = \left\langle {\left( {\overline{U} \,\,\,\,\overline{V} } \right)\,\,\left( {\begin{array}{*{20}c} {\tfrac{{\partial \overline{U} }}{\partial x}} & {\tfrac{{\partial \overline{V} }}{\partial x}} \\ {\tfrac{{\partial \overline{U} }}{\partial y}} & {\tfrac{{\partial \overline{V} }}{\partial y}} \\ \end{array} } \right)\,,\,\left( {\,\psi_{U} \,\,\psi_{V} } \right)_{i} } \right\rangle \hfill \\ e_{1i} = \int\limits_{\varOmega } {\left[ {\left( {\overline{U} \frac{{\partial \overline{U} }}{\partial x} + \overline{V} \frac{{\partial \overline{U} }}{\partial y}} \right)\psi_{Ui}^{{}} + \left( {\overline{U} \frac{{\partial \overline{V} }}{\partial x} + \overline{V} \frac{{\partial \overline{V} }}{\partial y}} \right)\psi_{Vi}^{{}} } \right]\,} dxdy \hfill \\ \end{gathered}$$

(A.8)

$$\begin{gathered} e_{2i} = \left\langle {\nabla \overline{{\mathbf{U}}} ,\nabla {\varvec{\uppsi}}_{i} } \right\rangle = \left\langle {\,\left( {\begin{array}{*{20}c} {\tfrac{{\partial \overline{U} }}{\partial x}} & {\tfrac{{\partial \overline{V} }}{\partial x}} \\ {\tfrac{{\partial \overline{U} }}{\partial y}} & {\tfrac{{\partial \overline{V} }}{\partial y}} \\ \end{array} } \right)\,,\,\,\left( {\begin{array}{*{20}c} {\tfrac{{\partial \psi_{U} }}{\partial x}} & {\tfrac{{\partial \psi_{V} }}{\partial x}} \\ {\tfrac{{\partial \psi_{U} }}{\partial y}} & {\tfrac{{\partial \psi_{V} }}{\partial y}} \\ \end{array} } \right)_{i} \,} \right\rangle \hfill \\ e_{2i} = \int\limits_{\varOmega } {\left[ {\frac{{\partial \overline{U} }}{\partial x}\frac{{\partial \psi_{Ui}^{{}} }}{\partial x} + \frac{{\partial \overline{U} }}{\partial y}\frac{{\partial \psi_{Ui}^{{}} }}{\partial y} + \frac{{\partial \overline{V} }}{\partial x}\frac{{\partial \psi_{Vi}^{{}} }}{\partial x} + \frac{{\partial \overline{V} }}{\partial y}\frac{{\partial \psi_{Vi}^{{}} }}{\partial y}} \right]\,} dxdy \hfill \\ \end{gathered}$$

(A.9)

It is also convenient to decompose the matrix A as $A_{ij} = - A_{1ij} - A_{2ij} - \tfrac{1}{Re}A_{3ij}$, with each term expanded as follows:

$$\begin{gathered} A_{1ij} = \left\langle {{\varvec{\uppsi}}_{j} \cdot \nabla \overline{{\mathbf{U}}} \,,\,{\varvec{\uppsi}}_{i} } \right\rangle = \left\langle {\left( {\,\psi_{U} \,\,\psi_{V} } \right)_{j} \,\,\left( {\begin{array}{*{20}c} {\tfrac{{\partial \overline{U} }}{\partial x}} & {\tfrac{{\partial \overline{V} }}{\partial x}} \\ {\tfrac{{\partial \overline{U} }}{\partial y}} & {\tfrac{{\partial \overline{V} }}{\partial y}} \\ \end{array} } \right)\,,\,\left( {\,\psi_{U} \,\,\psi_{V} } \right)_{i} } \right\rangle \hfill \\ A_{1ij} = \int\limits_{\varOmega } {\left[ {\left( {\psi_{Uj}^{{}} \frac{{\partial \overline{U} }}{\partial x} + \psi_{Vj}^{{}} \frac{{\partial \overline{U} }}{\partial y}} \right)\psi_{Ui}^{{}} + \left( {\psi_{Uj}^{{}} \frac{{\partial \overline{V} }}{\partial x} + \psi_{Vj}^{{}} \frac{{\partial \overline{V} }}{\partial y}} \right)\psi_{Vi}^{{}} } \right]\,} dxdy \hfill \\ \end{gathered}$$

(A.10)

$$\begin{gathered} A_{2ij} = \left\langle {\overline{{\mathbf{U}}} \cdot \nabla {\varvec{\uppsi}}_{j} \,,\,{\varvec{\uppsi}}_{i} } \right\rangle = \left\langle {\left( {\overline{U} \,\,\,\,\overline{V} } \right)\,\,\left( {\begin{array}{*{20}c} {\tfrac{{\partial \psi_{U} }}{\partial x}} & {\tfrac{{\partial \psi_{V} }}{\partial x}} \\ {\tfrac{{\partial \psi_{U} }}{\partial y}} & {\tfrac{{\partial \psi_{V} }}{\partial y}} \\ \end{array} } \right)_{j} \,,\,\left( {\,\psi_{U} \,\,\psi_{V} } \right)_{i} } \right\rangle \hfill \\ A_{2ij} = \int\limits_{\varOmega } {\left[ {\left( {\overline{U} \frac{{\partial \psi_{Uj}^{{}} }}{\partial x} + \overline{V} \frac{{\partial \psi_{Uj}^{{}} }}{\partial y}} \right)\psi_{Ui}^{{}} + \left( {\overline{U} \frac{{\partial \psi_{Vj}^{{}} }}{\partial x} + \overline{V} \frac{{\partial \psi_{Vj}^{{}} }}{\partial y}} \right)\psi_{Vi}^{{}} } \right]\,} dxdy \hfill \\ \end{gathered}$$

(A.11)

$$\begin{gathered} A_{3ij} = \left\langle {\nabla {\varvec{\uppsi}}_{j} ,\nabla {\varvec{\uppsi}}_{i} } \right\rangle = \left\langle {\,\,\left( {\begin{array}{*{20}c} {\tfrac{{\partial \psi_{U} }}{\partial x}} & {\tfrac{{\partial \psi_{V} }}{\partial x}} \\ {\tfrac{{\partial \psi_{U} }}{\partial y}} & {\tfrac{{\partial \psi_{V} }}{\partial y}} \\ \end{array} } \right)_{j} \,,\,\,\left( {\begin{array}{*{20}c} {\tfrac{{\partial \psi_{U} }}{\partial x}} & {\tfrac{{\partial \psi_{V} }}{\partial x}} \\ {\tfrac{{\partial \psi_{U} }}{\partial y}} & {\tfrac{{\partial \psi_{V} }}{\partial y}} \\ \end{array} } \right)_{i} \,} \right\rangle \hfill \\ A_{3ij} = \int\limits_{\varOmega } {\left[ {\frac{{\partial \psi_{Uj}^{{}} }}{\partial x}\frac{{\partial \psi_{Ui}^{{}} }}{\partial x} + \frac{{\partial \psi_{Uj}^{{}} }}{\partial y}\frac{{\partial \psi_{Ui}^{{}} }}{\partial y} + \frac{{\partial \psi_{Vj}^{{}} }}{\partial x}\frac{{\partial \psi_{Vi}^{{}} }}{\partial x} + \frac{{\partial \psi_{Vj}^{{}} }}{\partial y}\frac{{\partial \psi_{Vi}^{{}} }}{\partial y}} \right]\,} dxdy \hfill \\ \end{gathered}$$

(A.12)

Finally the third order tensor may be written as:

$$\begin{gathered} N_{ijk} = - \left\langle {{\varvec{\uppsi}}_{j} \cdot \nabla {\varvec{\uppsi}}_{k} ,{\varvec{\uppsi}}_{i} } \right\rangle = - \left\langle {\,\left( {\,\psi_{U} \,\,\psi_{V} } \right)_{j} \,\left( {\begin{array}{*{20}c} {\tfrac{{\partial \psi_{U} }}{\partial x}} & {\tfrac{{\partial \psi_{V} }}{\partial x}} \\ {\tfrac{{\partial \psi_{U} }}{\partial y}} & {\tfrac{{\partial \psi_{V} }}{\partial y}} \\ \end{array} } \right)_{k} \,,\,\,\,\left( {\,\psi_{U} \,\,\psi_{V} } \right)_{i} \,\,} \right\rangle \hfill \\ N_{ijk} = - \int\limits_{\varOmega } {\left[ {\left( {\psi_{Uj}^{{}} \frac{{\partial \psi_{Uk}^{{}} }}{\partial x} + \psi_{Vj}^{{}} \frac{{\partial \psi_{Uk}^{{}} }}{\partial y}} \right)\psi_{Ui}^{{}} + \left( {\psi_{Uj}^{{}} \frac{{\partial \psi_{Vk}^{{}} }}{\partial x} + \psi_{Vj}^{{}} \frac{{\partial \psi_{Vk}^{{}} }}{\partial y}} \right)\psi_{Vi}^{{}} } \right]\,} dxdy \hfill \\ \end{gathered}$$

(A.13)