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Reduced-order models have a number of practical engineering applications for unsteady flows that require either low-dimensional approximations for analysis and control or repeated simulation over a range of parameter values. The standard method for building reduced-order models uses the proper orthogonal decomposition (POD) and Galerkin projection. However, this standard method may be inaccurate when used “off-design” (at parameter values not used to generate the POD). This phenomena is exaggerated when parameter values describe the shape of the flow domain since slight changes in shape can have a significant influence on the flow field. In this paper, we investigate the use of POD sensitivity vectors to improve the accuracy and dynamical system properties of the reduced-order models to problems with shape parameters. To carry out this study, we consider flows past an elliptic cylinder with varying thickness ratios. Shape sensitivities (derivatives of flow variables with respect to thickness ratio) computed by finite-difference approximations are used to compute the POD sensitivity vectors. Numerical studies test the accuracy of the new bases to represent flow solutions over a range of parameter values.

Reduced-order modeling of incompressible flows plays an important role in academic and industrial research. In order to reduce the complexity of the governing equation, reduced-order models are often developed to represent the dynamical system with a few degrees of freedom. These models can provide analytical insight into the physical phenomena and enable application of dynamical systems theory and control methods.

Since Roshko [

On the other hand, engineering applications often involve flows over complex bodies, such as wings, submarines, missiles, and rotor blades, which can hardly be modeled as a circular cylinder. It would however be interesting to broaden the application of reduced-order models to other complex geometries. A systematic approach would be to start by studying a configuration that is more general than a circular cylinder and can characterize typical engineering flow configurations. Elliptic cylinders seem to provide such a configuration and changes in the eccentricity allow for shapes ranging from a circular cylinder to a flat plate. By tuning the damping coefficients, self-excited oscillator models [

In some cases, the reduced-order models represent the space variation with a limited number of basis functions, while still capturing the physics and dominant features of the flow. Most widely used reduced-order model techniques in fluid dynamics are derived from the proper orthogonal decomposition- (POD-) Galerkin projection approach [

Despite the accurate reproduction of the data from which it is originated, the POD-based reduced-order model lacks robustness away from the

Various methods have been proposed to extend the applicability and increase the accuracy of POD-based reduced-order models away from the

In the application of reduced-order models to optimal control, one desires a model that is accurate over a wide range of actuation data. In a recent study, Graham et al. [

In our previous work [

A follow-up study [

In this study, we use a similar basis improvement methodology. However, we consider modeling the flow as the obstacle is changed from a circular cylinder to an elliptic cylinder. The shape parameter is chosen as the thickness ratio (

The paper is organized as follows. Section

The Navier-Stokes and continuity equations are the governing equations for the present problem. For incompressible flow, they can be represented as follows:

We employ curvilinear coordinates in an Eulerian reference frame; a planar view is shown in Figure

(a) Geometry of an elliptic cylinder and (b) “O” grid distributed among 8 processors in the

The Navier-Stokes equations are written in curvilinear coordinates and strong conservation form as follows:

In this study, a body conformal O-type grid is employed to simulate the flow over a body as shown in Figure

A semi-implicit scheme is employed to advance the solution in time. The diagonal viscous terms are advanced implicitly using the second-order accurate Crank-Nicolson method; whereas all of the other terms are advanced using the second-order accurate Adams-Bashforth method. In the present formulation, we apply a fractional-step method to advance the solution in time. The fractional-step method splits the momentum equation into

an advection-diffusion equation—the momentum equation solved without the pressure term,

a pressure Poisson equation—constructed by implicit coupling between the continuity equation and the pressure in the momentum equation, thus satisfying the constraint of mass conservation.

The governing equations are solved using a methodology similar to that employed by Zang et al. [

The CFD results for the numerical methodology have been validated for three-dimensional [

Mathematically, we compute

In the classical POD or direct method, originally introduced by Bakewell and Lumley [

The flow data or

We write the velocity field as the sum of the mean flow (

Substituting (

We perform numerical simulations of the flows past elliptic cylinders with thickness ratios ranging from

We develop the reduced-order model for the flow past a circular cylinder (as an example) and use a similar procedure for the model reduction of the flow past elliptic cylinders with different thickness ratios. In the current study, we took 64 snapshots of the flow field over one vortex shedding cycle of the circular cylinder. The flow data or

Streamwise POD modes for

In this section, we derive the first-order total derivatives of the POD modes with respect to a generic shape parameter

The sensitivities with respect to the

The traditional approach in reduced-order modeling is to build the POD basis for one particular value of the parameter of the system. This will be referred to as the baseline approach and denoted by

Following the previous studies for flow parameters in [

In the current study, we numerically simulate the flow past elliptic cylinders with varying thickness ratios ranging from

Instantaneous spanwise vorticity contours.

The velocity coefficients

From a reduced-order modeling point of view, variation in any geometric parameter would require a new set of flow data or snapshots, computation of the POD modes, and the projection onto these modes to develop a new model. This procedure corresponds to using the

Using the sensitivity analysis, we can modify the POD basis to include the effect of parameter variation in the flow field. To do so, we consider the thickness ratio of the ellipse

Two-dimensional projections of the phase portraits for different bases.

Relative error in reduced-order models at

We vary

Relative error in reduced-order approximations.

In general, the expanded basis tends to perform better than extrapolated basis. Since we assume linear dependency of the POD modes on the geometric parameter, finite-difference approach makes the extrapolated approach close to the baseline methodology. In general, the model based on expanded basis shows relatively better results close to

We investigated the possibility of using the POD sensitivity vectors corresponding to a change in shape to improve the accuracy and dynamical system properties of the reduced-order models. As a part of the ongoing research in this area, we modified a circular cylinder to an elliptic cylinder by changing its thickness ratio and computed the sensitivity in the POD modes with respect to this thickness ratio. We defined different POD bases functions, with and without sensitivity, and used them to approximate the velocity field. We then compared the performance of these bases and found that the inclusion of shape sensitivity information in the POD bases performs better than the baseline approach. However, the Sensitivity Analysis for shape parameters is more challenging than for flow parameters since the extrapolation explicitly requires the mapping from one domain to another. The results from the shape sensitivity are encouraging and require further investigation in this field, especially when the parameter changes lead to bifurcations which would require higher order sensitivities (and not merely the first order derivatives).

The POD eigenfunctions are used as a basis in a Galerkin projection of the incompressible Navier-Stokes equations. The projection is performed by the inner product of the POD modes with (

From the Galerkin expansion in (

We substitute (

The pressure term in the model is also projected onto the POD modes as follows:

We substitute (

This research was partially supported by the Air Force Office of Scientific Research under Contract FA9550-08-1-0136. Numerical simulations were performed on Virginia Tech Advanced Research Computing—System X. The allocation grant and support provided by the staff are also gratefully acknowledged. Imran Akhtar would like to thank the Government of Pakistan for support during his graduate studies.