Reduced order models for thermally coupled low Mach flows
Abstract
In this paper we present a collection of techniques used to formulate a projectionbased reduced order model (ROM) for zero Mach limit thermally coupled Navier–Stokes equations. The formulation derives from a standard proper orthogonal decomposition (POD) model reduction, and includes modifications to improve the drawbacks caused by the inherent nonlinearity of the used Navier–Stokes equations: a hyperROM technique based on mesh coarsening; an implicit ROM subscales formulation based on a variational multiscale (VMS) framework; and a Petrov–Galerkin projection necessary in the case of nonsymmetric terms. At the end of the article, we test the proposed ROM formulation using 2D and 3D versions of the same example: a differentially heated cavity.
Keywords
Reduced order models Finite element method Variational multiscale Hyperreduction Thermally coupled flow Low Mach number flowIntroduction
The main purpose of this paper is to develop a model reduction formulation suitable for thermally coupled flows, by expanding the techniques in projectionbased model reduction developed for several applications on fluid dynamics—mostly for incompressible Navier–Stokes equations—to the zero Mach limit Navier–Stokes equations developed in [1, 2].
Following the model reduction developments in [3] we choose a POD model reduction approach. The POD, as any projectionbased model reduction, aims to describe any phenomena that otherwise would be represented by a ‘computationally expensive’ numerical method with a surrogate lower dimensional model. This surrogate model is obtained by projecting the computational expensive numerical approximation onto a previously computed reduced space. Thus, the model reduction is arranged in two stages: an offline part, where the solution obtained from a ‘high fidelity’ full order model (FOM) is used to build the desired reduced order subspace; and an online part, where by projecting the original model onto the reduced subspace the ROM is built and subsequently solved.
The traditional POD model reduction approach presents certain drawbacks when considering nonlinear complex problems: added computational cost of representing the nonlinearities over a linearized computational model, inherent numerical instabilities caused by the Navier–Stokes formulation, and nonoptimal projection over the reduced space caused by the asymmetrical nature of the formulation.
To overcome the first issue, a wide variety of methods inspired by the work of Everson and Sirovich [4] have been introduced [5, 6, 7, 8, 9, 10], with the term ‘hyperROM’ coined by Ryckelink [11]. This family of methods consists in using a sample of the geometrical domain. In this paper, we propose an idea of hyperROM that differs from the sampling way: to set the ROM on a geometrical space smaller than the original one. This, applied to mesh based methods, implies the interpolation of the developed ROM—basis included—onto a coarser mesh.
To stabilize both the FOM and ROM, we follow the same approach: a VMS framework. For the FOM case, we follow the stabilized formulation of the thermally coupled zero Mach limit Navier–Stokes equations developed in [12], using the Orthogonal SubgridScales (OSS) as defined in [13]. For the ROM case, we propose an analogous method where the OSS are defined orthogonal to the ROM subspace.
Lastly, to solve the nonoptimality in the reduced space projection, we use a Petrov–Galerkin projection instead of the standard Galerkin projection as proposed in [7]. In the specific case of mesh based methods, the PODROM method can be seen as a solution of the original problem using a projection method as in [14], where the Petrov–Galerkin projection is the one that satisfies the nonsingularity of the system of equations when the FOM is not symmetric.
This article is organized as follows. In the first section, we present a brief description of the zero Mach number limit Navier–Stokes model. In the second section, we describe briefly the idea of model reduction along with an explanation of the POD compression technique for the construction of the basis. In the third section, we present a modified finite element (FE)ROM implementation, including the stabilization using VMS, the new proposed hyperROM method, and the Petrov–Galerkin projection. In the fourth section, we present examples consisting in a differential heated cavity, with 2D and 3D cases. Finally, we close the paper with some conclusions.
Thermally coupled flow problems
Low Mach number model
Model reduction
Projectionbased ROMs rely on the existence of a reduced dimensional subspace that approximates the solution space. Let us define a high dimensional space Open image in new window of dimension M, with \(\varvec{\varphi }\) its orthonormal basis. Then, the ith component of any element Open image in new window can be written as the linear combination \(Y_{h,i} = \sum \nolimits _{k=1}^M (Y_{h,i},\varphi ^k_i) \varphi ^k_i\), with ( : , : ) a \(L^2\)inner product. Note that Open image in new window may be an approximation to a continuous—infinite dimensional—space. Since for most cases the exact basis \(\varvec{\varphi }\) is unknown, we can define a lowerdimensional space Open image in new window of dimension m, which approximates Open image in new window as \(m \rightarrow M\), with a basis \(\varvec{\phi }\). Using this test basis, we can approximate the ith component of any element \(\varvec{Y}_h\) as \(Y_{h,i} \approx \sum \nolimits _{k=1}^m \phi ^k_i Y^k\), where the accuracy of the approximation will depend on how accurate is the basis \(\varvec{\phi }\) compared to the exact basis \(\varvec{\varphi }\).
Construction of the basis
Following previous works in model reduction in FE [3, 7, 9], we use the POD statistical procedure as a way to build the reduced order subspace basis \(\varvec{\phi }\). The objective of the POD method is finding a basis for a collection of highfidelity ‘snapshots’ to use it as a the basis of the desired reduced subspace.
Remark
Note that this is the way we approximate \(\varvec{\phi }\), but there are several other ways to find a basis for the reduced order subspace (as the ones in [15, 16]). The projectionbased model reduction formulation below should be valid for any basis regardless of the technique used to obtain it.
Remark
In this paper, the ‘snapshots’ data is obtained using a FE approximation and therefore, the basis \(\varvec{\phi }\) approximates the FE space Open image in new window —of dimension M—which in turn approximates the space of the continuous problem.
FEROM formulation
Having stated the equations that represent the physical problem and the standard model reduction approach, we now describe the FEROM approximation of the problem.
In this section, let us denote Open image in new window as the functional continuous space where \(\varvec{Y}\) exist. Instead of following a standard Galerkin approximation of the variational problem—where the FE space is denoted as Open image in new window —we construct the approximation space Open image in new window for the ROM. The FE space is assumed to be built from a FE partition Open image in new window of the domain \(\Omega \). The order of the FE interpolation is irrelevant for our discussion.
VMS for FEROM
Given the wellknown lack of stability in the Galerkin standard formulation—present in convective dominated regimes—a stabilization technique is necessary. Inspired in previous works that acknowledge instability issues [17, 18] and using a VMS framework as done for several problems in FE approximations, we develop what we call FEROMSubgridScales (SGS), which resembles FESGS.
Subscales approximation
Remark
By following the same analysis performed when deriving the orthogonal SGS in [13, 19], we have come to a rather similar definition of the FEROMSGS, where the most important difference lies in the definition of the orthogonal projection \(\Pi ^\perp \) in Eq. 26. In the FE case the projection is done onto the space Open image in new window , while in the FEROM approximation it is done onto the space Open image in new window .
Remark
The choice of the stabilization parameters \(\varvec{\tau }\) is done following the Fourier analysis done in [13]. Since the information represented by the reduced basis corresponds to the resolved scales from the FOM, the subscales for both the FOM and the ROM are part of the continuous solution which cannot be approximated by the FOM.
Remark
The previous definition of the FEROMSGS is equivalent to the dynamic orthogonal SGS model in [20]; it is important to acknowledge that subscales could also be implemented without the temporal term (quasistatic), or not orthogonal to Open image in new window . An extensive analysis of the FE equivalent models is depicted in [12, 20].
Time discretization
Linearization
To solve the nonlinearity present in the terms involving \(\varvec{Y}\), we implement a linearization scheme based on Picard’s method. Using the terminology used in [20], for each time step \(n+1\), we first solve Eq. 30 for iteration \(i+1\), where the nonlinear terms can be approximated in two ways: as \(\varvec{Y}_r^{n+1,i}\), for linear subscales; or as \(\varvec{Y}_r^{n+1,i} + \breve{\varvec{Y}}^{n+1,i}\), for nonlinear subscales. Then we solve Eq. 29 for iteration \(j+1\), approximating the nonlinear terms in the same way: by linear subscales (\(\varvec{Y}_r^{n+1,i+1}\)); or nonlinear subscales (\(\varvec{Y}_r^{n+1,i+1} + \breve{\varvec{Y}}^{n+1,j}\)).
Discrete approximation
We can describe the discrete representation of the FEROM problem as a composition of the FE and ROM approximations. In FE the space Open image in new window is defined as made of continuous piecewise polynomial functions in the domain \(\Omega \), where we can write the discrete approximation of the unknown as \(\varvec{Y} \approx \varvec{Y}_h(\varvec{x},t) {:}{=}\sum \nolimits _{i=1}^n N(\varvec{x}^i) \varvec{Y}^{i}(t)\), with \(N(\varvec{x}^i)\) the shape function of node i. In contrast, in ROM we approximate the unknown \(\varvec{Y}\) as \(\varvec{Y}(t) \approx \bar{\varvec{Y}} + \sum \nolimits _{k=1}^m \varvec{\phi }^k Y^k (t)\).
Petrov–Galerkin projection
HyperROM
Lastly, in order to reduce the computational cost of evaluating nonlinear terms, we propose a meshbased hyperROM as an alternative to the samplingbased domain reduction algorithms [5, 6, 7, 8, 9, 10].
The meshbased hyperROM consists in the solution of the described ROM problem using a coarser mesh than the one of the FOM. The implementation of this technique is done straightforwardly by writing the discrete approximation (Eq. 31) in function of the new coarser mesh.
Ideally, the coarsening should be performed as a function of the ‘less important’ areas of the geometry, which can be achieved using already existing mesh refinement algorithms. In the subsequent examples we test this technique using a uniform coarsening of the mesh.
Remark
When the POD basis is obtained by sampling a meshbased solution—a FE one for example—the coarsening of the mesh implies an interpolation of such basis.
Numerical examples
In this section we present two examples consisting of 2D and 3D versions of the initial transient part of a differentially heated cavity of aspect ratio 1—similar to the one presented in [12, 22]. In both examples the flow is considered an ideal gas with physical properties \(R=287.0 \frac{\text {J}}{\text {kg}\text {K}}\), \(c_p=1004.5 \frac{\text {J}}{\text {kg}\text {K}}\), and \(\mu =0.001 \frac{\text {kg}}{\text {m s}}\).
The computational domain for both problems is defined as \(\Omega = [0,L] \times [0,L]\) for the 2D problem, and \(\Omega = [0,L] \times [0,L] \times [0,L]\) for the 3D problem, with \(L= 1\) m. The temperatures on the walls perpendicular to the xcoordinate (horizontal) are fixed to \(T_h = 960\) K and \(T_c= 240\) K; while adiabatic boundary conditions are prescribed in the remaining walls. Additionally, no slip and impermeable conditions are set over all the walls, together with a homogeneous gravity force \(\varvec{g}\) prescribed in the negative ycoordinate (vertical). The initial thermodynamic pressure, temperature and density are set to \(p^{th}_0=101,325\) Pa, \(T_0 = 600\) K, and \(\rho _0 =0.58841 \frac{\text {kg}}{\text {m}^3}\) respectively, and the dimensionless Prandtl and Rayleigh numbers are set to \(Pr = \frac{c_p \mu }{\lambda } = 0.71\) and \(Ra = \frac{ 2\varvec{g} \rho ^2 c_p}{\lambda \mu } \frac{(T_h  T_c)}{(T_h + T_c)} = 3.55 \cdot 10^{6}\).
For the FOM solution (as a reference solution and for the construction of the snapshots), we follow the VMS formulation presented in [20], using the dynamic orthogonal SGS model. In all cases, we use a constant time step size of \(\delta t = 0.01\) s. For the finite element meshes used (described below) this time step is slightly higher than the critical time step of an explicit scheme due to advection, which in turn is of the same order as that due to viscous effects. Since we use an implicit scheme, we are not restricted to a critical time step size, but this observation serves to justify that our choice is adequate.
Additionally to the following examples, we have tested a differentially heated cavity of aspect ratio 8 analyzed in [23], getting inconclusive results.
Two dimensional case
As expected, we observe a more diffusive behaviour, in both the mean and the fluctuation of the Nusselt number, when fewer basis vectors are included. The hyperROM results appear to have the same behaviour of the ROM with lower amplitude. Additionally, we include a FOM solution (labeled \(\text {FOM}_{H}\)) using the coarser mesh to evaluate how the hyperROM formulation relates to a FE mesh coarsening.
Although the convergence error does not have a clear slope, it behaves as expected, with the error decreasing with the addition of basis vectors. It is important to notice that appears to be an optimal value for \(\eta \ne 1\)—the maximum number of basis vectors—where the error reaches the minimum; considering that this results occur near \(\eta =1\), we believe it can attributed to an overfitting phenomenon. The same idea is explored in [24, 25], where it is attributed to a lack of smoothness or noisiness in low energy basis vectors. The overall results for the hyperROM seem adequate, given that the error is the same order of the ROM and the coarse mesh FOM solutions.
Again we see how reducing the amount of basis vectors leads to a more diffusive solution. But in contrast to Fig. 7, in Nusselt number spectra we observe that the ROM and hyperROM spectra tend to the FOM spectra as we approach \(\eta =1\).
Three dimensional case
For the 3D problem, we use 2 uniform structured meshes composed of regular hexahedral elements: one with 64,000 elements and a mesh size \(h=0.025\), used for the solving the FOM and the ROM; and one with 35937 elements and a mesh size \(h=0.\bar{03}\), for solving the hyperROM. To construct the basis, we collect 500 snapshots for velocity, pressure and temperature at every 2 time steps in a 10 s interval.
As in the 2D problem, for the 3D case we compute the Nusselt number, the root mean square of the Nusselt error and the discrete Fourier transform of the Nusselt number for the hot and cold walls, getting similar results.
Conclusions
In this work we have developed a formulation that allowed us to perform ROMs on the zero Mach number Navier–Stokes approximation. For that purpose we have described a set of tools that allow us to tackle the main problems that arise: lack of stability, added computational cost due to the nonlinearities and nonoptimal projection over the reduced space.

It is an implicit formulation, reducing the offline costs and easing the implementation.

Since it is built over a VMS framework, it incorporates nonlinear terms in the adjoint operator (Eq. 15), which may become relevant in complex flows.

It maintains the orthogonality definition, proposed in [18, 26], between the subspaces Open image in new window and Open image in new window .

It is residualbased, as shown in Eq. 18.

It works with the same stabilization coefficients \(\varvec{\tau }\) as the ones developed for the VMSFE formulation in [2, 12].
To solve the added computational cost, we have proposed a meshbased hyperROM that contrary to the traditional samplebased methods, does not require any algorithm to select these sampling points. The numerical experiments performed show that this hyperROM method behaves appropriately, only inducing an expected added diffusion to the solution. A natural extension of the meshbased hyperROM method is the use of mesh refinement techniques to improve both the computational time and the accuracy of the method.
Finally, to tackle the nonoptimal projection of the nonlinear formulation, we give a different interpretation to the Petrov–Galerkin projection—originally described in [7]—in order to include it in our formulation.
Notes
Author's contributions
All authors contributed to the manuscript both in writing and the numerical experiments. The computer code for the numerical simulations was developed in FEMUSS by RR. All authors read and approved the final manuscript.
Acknowledgements
R. Reyes acknowledges the scholarship received from COLCIENCIAS, from the Colombian Government. R. Codina acknowledges the support received from the ICREA Academia Research Program, from the Catalan Government.
Competing interests
The authors declare that they have no competing interests.
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