Iterative Brinkman penalization for remeshed vortex methods

doi:10.1016/j.jcp.2014.09.029

Journal of Computational Physics

Volume 280, 1 January 2015, Pages 547-562

https://doi.org/10.1016/j.jcp.2014.09.029 Get rights and content

Abstract

We introduce an iterative Brinkman penalization method for the enforcement of the no-slip boundary condition in remeshed vortex methods. In the proposed method, the Brinkman penalization is applied iteratively only in the neighborhood of the body. This allows for using significantly larger time steps, than what is customary in the Brinkman penalization, thus reducing its computational cost while maintaining the capability of the method to handle complex geometries. We demonstrate the accuracy of our method by considering challenging benchmark problems such as flow past an impulsively started cylinder and normal to an impulsively started and accelerated flat plate. We find that the present method enhances significantly the accuracy of the Brinkman penalization technique for the simulations of highly unsteady flows past complex geometries.

Introduction

Direct Numerical simulations of bluff body flows using remeshed vortex methods have been traditionally associated with the computation of the vorticity flux, via boundary element methods (BEM), for the enforcement of the no-slip boundary condition [1]. In this context, the no-slip boundary condition is enforced in a fractional step algorithm. First, the no-through flow boundary condition is enforced by solving the potential flow problem using a vortex sheet, which is equivalent to a tangential slip velocity at the solid surface. The vortex sheet is then translated into a vorticity flux in order to enforce the no-slip boundary condition [2].

The particle velocity can be computed by solving the Poisson equation [1], [3], [4] using the Fast Multipole [5] or mesh based method (as in vortex-in-cell algorithms [6]). A combination of the boundary element method and the vortex-in-cell [6] method was presented in [7] using a particle–particle particle–mesh ( $P^{3} M$ ) algorithm [8] to correct the vortex-in-cell method with the vorticity obtained from the BEM.

The distortion of the particle locations can render vortex methods inaccurate. In order to remedy this problem, remeshing was introduced [1], [9] to regularize the particle locations. In remeshed vortex methods particles are advected with the velocity of the flow field. When particle locations get distorted their strength is interpolated onto mesh nodes, which become the particles to be advected in the next iteration. The Poisson equation as well as the viscous terms can be computed during remeshing [10] or after the remeshing step, using finite difference operators.

In remeshing, particle–mesh interpolations are computationally efficient by employing Cartesian grids [9] and interpolation functions that are tensorial products. The drawback of using a Cartesian mesh is that during remeshing spurious vorticity can be introduced in the interior of the body. A remedy is to use locally one-sided interpolation or body fitted grids [1], [11], [12]. However both of these approaches have limited applicability. In order to overcome this difficulty for complex, deforming geometries, in recent years a number of efforts have combined the Brinkman penalization techniques [13], [14] with remeshed vortex methods [15], [16], [17], [18], [19]. These simulations have demonstrated the capability of the penalization method to produce results in good agreement with benchmark simulations. A major drawback of these methods however, is that they impose a stringent time step on advancing the flow field to accurately capture vorticity generation at the boundary.

Here we propose a remedy to this situation by using an iterative scheme that ensures the accurate generation of vorticity at the boundary within each time step of the penalty method. We demonstrate the improvement in accuracy over the classical method, using a number of challenging benchmark problems, such as flow past an impulsively started cylinder and normal to a thin flat plate.

The paper is structured as follows: We introduce the governing equations and numerical discretization in Sections 2 Governing equations and numerical method, 3 Brinkman penalization of incompressible viscous flow, 4 An iterative Brinkman penalization method for incompressible viscous flow, followed by simulations (Sections 5 Impulsively started flow past a circular cylinder at, 6 Impulsively started flow normal to a flat plate of finite thickness at, 7 Flow past an impulsively started plate, inclined at 45° and, 8 Uniformly accelerated flow normal to a flat plate of finite thickness at an acceleration-formulated Reynolds number of) of the flow past an impulsively started cylinder at $Re = 9500$ , the impulsively started flow normal to a flat plate at $Re = 1000$ , the impulsively started flow at a 45 degree angle to a flat plate at $Re = 1000$ , and the flow normal to a uniformly accelerated flat plate with an acceleration-based $Re = 16.8 \times 10^{5}$ . We conclude by discussing the advantages of the present method over the classical penalization techniques.

Section snippets

Governing equations and numerical method

The velocity-vorticity form of the two-dimensional Navier–Stokes equations can be expressed in a Lagrangian frame as: $\frac{d x}{d t} = u and \frac{D ω}{D t} = ν \nabla^{2} ω .$ where ν is the kinematic viscosity, and the vorticity ω is defined as the curl of the velocity field $ω = \nabla \times u$ . The velocity field u can be represented by a Helmholtz decomposition $u = \nabla \times ψ + \nabla ϕ$ where ψ is the stream function and ϕ is the velocity potential. For an incompressible flow ( $\nabla \cdot u = 0$ ) the stream function can be obtained by solving a Poisson equation: $\nabla^{2} u = - \nabla \times ω .$

Brinkman penalization of incompressible viscous flow

The penalization method enforces the no-slip boundary condition on the surface of a body in an incompressible flow [13] by introducing a source term localized around the surface of the body. The velocity of the flow u is modified by the penalization term as: $\frac{\partial u}{\partial t} = λ [χ (v - \tilde{u})] .$ where v denotes the velocity of the body and $\tilde{u}$ denotes the velocity field of the flow prior to penalization i.e. from the solution of Eq. (3). The penalization parameter λ has units of reciprocal time and is equivalent to a

An iterative Brinkman penalization method for incompressible viscous flow

We remark that while the velocity field is related to the vorticity through the elliptic Poisson equation, the penalization introduces only a local relation between velocity and vorticity. This inconsistency can be attributed to the non-consistent handling of the flow kinematics and the no-through flow boundary condition by the classical Brinkman penalization technique.

Here we propose to compute the vorticity field from the penalty term and use it in turn to compute an updated velocity field,

Impulsively started flow past a circular cylinder at $Re = 9500$

In this section we compare the new iterative penalization with the classical non-iterative explicit penalization method for the benchmark case of the impulsively started flow past a circular cylinder at $Re = 9500$ . The Reynolds number, the non-dimensional time, and the drag coefficient for this case are defined by $Re = \frac{D U_{\infty}}{ν}, t^{⁎} = \frac{t U_{\infty}}{D}, and C_{D} = \frac{2 F \cdot e_{x}}{U_{\infty} D}$ where D is the diameter of the cylinder. In order to compare with the BEM based vortex method of Koumoutsakos and Leonard [1] the simulation is performed

Impulsively started flow normal to a flat plate of finite thickness at $Re = 1000$

As mentioned earlier, the penalization method alone does not enforce a global correction of the flow field. A consequence of this is an inherent delay in producing proper surface vorticity to enforce the no-through boundary condition. Hence the flow normal to a flat plate presents a particularly difficult test case for the penalization method. For the impulsively started flow normal to a flat plate the Reynolds number, non-dimensional time, non-dimensional plate thickness, and the drag

Flow past an impulsively started plate, inclined at 45° and $Re = 1000$

In order to validate the method on an inherently asymmetrical flow, we study the flow past an impulsively started plate at 45° with a thickness $H^{⁎} = 1 / 50$ and $Re = 1000$ . The non-dimensionalized parameters are the same as those stated in Section 6 cf. Eq. (25). In the present test case, the spatial resolution $δ x / L = 1 / 400$ , corresponding to 8 cell lengths across the plate thickness, and the time step is $δ t^{⁎} = 10^{- 3}$ .

We observe in Fig. 11 that the penalization method does not conserve the zeroth moment of

Uniformly accelerated flow normal to a flat plate of finite thickness at an acceleration-formulated Reynolds number of $16.8 \times 10^{5}$

In this section we investigate a uniformly accelerated flow normal to a flat plate of finite thickness. The acceleration normalized Reynolds number β, the non-dimensional time, and the drag coefficient as given for this case by Koumoutsakos and Shiels [11] are: $β = \frac{a L^{3}}{ν^{2}}, t^{⁎} = \frac{a t^{2}}{L}, and C_{D} = \frac{2 F \cdot e_{x}}{{(a t_{f})}^{2} L},$ where a is the constant acceleration of the plate.

Following [11] we chose $t_{f}$ such that $t_{f}^{⁎} = 25$ , and $a = 1.68$ and $ν = 10^{- 3}$ to reach an acceleration-formulated Reynolds number of $β = 16.8 \times 10^{5}$ . The time step used

Conclusion

We demonstrate the need to extend the Brinkman penalization technique, by performing sub-iterations so as to enhance its accuracy in enforcing the no-through as well as the no-slip flow boundary conditions.

We have coupled this iterative Brinkman penalization technique with remeshed vortex methods. We demonstrate the improved accuracy of our method using benchmark problems computed by remeshed vortex methods with and without penalization. The calculated flow induced forces are in excellent

Acknowledgement

We would like to acknowledge the helpful discussions with Henrik Juul Spietz, Johannes Tophøj Rasmussen, Babak Hejazialhosseini and Wim van Rees. We are grateful for the lift and drag data provided by Jeff D. Eldredge. Finally, we thank the reviewers for their insightful and helpful comments.

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Iterative Brinkman penalization for remeshed vortex methods

Abstract

Introduction

Section snippets

Governing equations and numerical method

Brinkman penalization of incompressible viscous flow

An iterative Brinkman penalization method for incompressible viscous flow

Impulsively started flow past a circular cylinder at Re=9500

Impulsively started flow normal to a flat plate of finite thickness at Re=1000

Flow past an impulsively started plate, inclined at 45° and Re=1000

Uniformly accelerated flow normal to a flat plate of finite thickness at an acceleration-formulated Reynolds number of 16.8×105

Conclusion

Acknowledgement

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Comput. Struct.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Eur. J. Mech. B, Fluids

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Impulsively started flow past a circular cylinder at $Re = 9500$

Impulsively started flow normal to a flat plate of finite thickness at $Re = 1000$

Flow past an impulsively started plate, inclined at 45° and $Re = 1000$

Uniformly accelerated flow normal to a flat plate of finite thickness at an acceleration-formulated Reynolds number of $16.8 \times 10^{5}$