A multirate time integrator for regularized Stokeslets

The method of regularized Stokeslets is a numerical approach to approximating solutions of fluid–structure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized point-force term that are used to represent forces generated by a rigid or elastic object interacting with the fluid. Due to the linearity of the Stokes equations, the velocity at any point in the fluid can be computed by summing the contributions of regularized Stokeslets, and the time evolution of positions can be computed using standard methods for ordinary differential equations. Rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized Stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid. For problems with boundaries modeled by springs with large spring constants, the resulting ordinary differential equations become stiff, and hence the time step for explicit time integration methods is severely constrained. Unfortunately, the use of standard implicit time integration methods for the method of regularized Stokeslets requires the solution of dense nonlinear systems of equations for many relevant problems. Here, an alternate strategy using an explicit multirate time integration scheme based on spectral deferred corrections is incorporated that in many cases can significantly decrease the computational cost of the method. The multirate methods are higher-order methods that treat different portions of the ODE explicitly with different time steps depending on the stiffness of each component. Numerical examples on two nontrivial three-dimensional problems demonstrate the increased efficiency of the multi-explicit approach with no significant increase in numerical error.     

The method of images for regularized Stokeslets

The image system for the method of regularized Stokeslets is developed and implemented. The method uses smooth localized functions to approximate a delta distribution in the derivation of the fluid flow due to a concentrated force. In order to satisfy zero-flow boundary conditions at a plane wall, the method of images derived for a standard (singular) Stokeslet is extended to give exact cancellation of the regularized flow at the wall. As the regularization parameter vanishes, the expressions reduce to the known images for singular Stokeslets. The advantage of the regularized method is that it gives bounded velocity fields even for isolated forces or for distributions of forces along curves. These are useful in the simulation of ciliary beats, flagellar motion, and particle suspensions. The expression relating force and velocity can be inverted to find the forces that generate a given velocity boundary condition. The latter is exemplified by modeling a cilium as a filament moving in a three-dimensional flow. The cilium velocity at various times is constructed from known data and used to determine the force field along the filament. Those forces can then reproduce the flow everywhere. The validity of the method is evaluated by computing the drag on a sphere moving near a wall. Comparisons with known expressions for the drag show that the method gives accurate results for spheres even within a distance from the wall equal to the surface discretization size.     

A note on pressure accuracy in immersed boundary method for Stokes flow

In this short note, we provide a simplified one-dimensional analysis and two-dimensional numerical experiments to predict that the overall accuracy for the pressure or indicator function in immersed boundary calculations is first-order accurate in L₁ norm, half-order accurate in L₂ norm, but has O(1) error in L_∞ norm. Despite the pressure has O(1) error near the interface, the velocity field still has the first-order convergence in immersed boundary calculations.     

Towards oscillation-free implementation of the immersed boundary method with spectral-like methods

It is known that, when the immersed boundary method (IBM) is implemented within spectral-like methods, the Gibbs oscillation seriously deteriorates the calculation of derivatives near the body surface. In this paper, a radial basis function (RBF) based smoothing technique is proposed with the intention of eliminating or efficiently reducing the Gibbs oscillation without affecting the flow field outside the body. Based on this technique, a combined IBM/spectral scheme is developed to solve the incompressible Navier–Stokes equations. Numerical simulations of flow through a periodic lattice of cylinders of various cross sections are performed. The results demonstrate that the proposed methodology is able to give accurate and nearly oscillation-free numerical solutions of incompressible viscous flows.     

Incorporation of smooth spherical bodies in the Lattice Boltzmann method

A method to simulate bodies suspended in a Lattice Boltzmann solvent is proposed. It is based on a generalized reaction force that enforces no-slip boundary conditions at the fluid–body interface as the limiting case of an iterative procedure. A smooth version of the Heaviside function allows to treat spherical particles of arbitrary size and produces smooth hydrodynamic forces as particles move in the continuum. Numerical tests demonstrate the accuracy of the method in reproducing the hydrodynamic field around a single particle and the fluid-mediated forces between pairs of particles. The drag force experienced by a particle moving in a straight channel and at various Reynolds numbers is studied as a non-trivial testcase.     

An efficient multi-scale Poisson solver for the incompressible Navier–Stokes equations with immersed boundaries

The iterative-multi-scale-finite-volume (IMSFV) procedure is applied as an efficient solver for the pressure Poisson equation arising in numerical methods for the simulation of incompressible flows with the immersed-interface method (IIM). Motivated by the requirements of the specific IIM implementation, a modified version of the IMSFV algorithm is presented to allow the solution of problems, in which the varying coefficient of the elliptic equation (e.g. the permeability of the medium in the context of the simulation of flows in porous media) varies over several orders of magnitude or even becomes zero within the integration domain. Furthermore, a strategy is proposed to incorporate the iterative procedure needed by the IIM to converge out constraints at immersed boundaries into the iterative IMSFV cycle. No significant deterioration of performance of the IMSFV method is observed with respect to cases, in which no iterative improvement of the boundary conditions is considered.     

A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations

The effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods. The problem with this representation is that the volume force exhibits non-physical oscillations in moving boundary simulations. A smoothing technique for discrete delta functions has been developed in this paper to suppress the non-physical oscillations in the volume forces. We have found that the non-physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions. It has been shown that the smoothed discrete delta functions constructed in this paper have one-order higher derivative than the regular ones. Moreover, not only the smoothed discrete delta functions satisfy the first two discrete moment conditions, but also their derivatives satisfy one-order higher moment condition than the regular ones. The smoothed discrete delta functions are tested by three test cases: a one-dimensional heat equation with a moving singular force, a two-dimensional flow past an oscillating cylinder, and the vortex-induced vibration of a cylinder. The numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non-physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations.     

Prediction of wall-pressure fluctuation in turbulent flows with an immersed boundary method

The objective of this paper is to assess the accuracy and efficiency of the immersed boundary (IB) method to predict the wall pressure fluctuations in turbulent flows, where the flow dynamics in the near-wall region is fundamental to correctly predict the overall flow. The present approach achieves sufficient accuracy at the immersed boundary and overcomes deficiencies in previous IB methods by introducing additional constraints – a compatibility for the interpolated velocity boundary condition related to mass conservation and the formal decoupling of the pressure on this surfaces. The immersed boundary-approximated domain method (IB-ADM) developed in the present study satisfies these conditions with an inexpensive computational overhead. The IB-ADM correctly predicts the near-wall velocity, pressure and scalar fields in several example problems, including flows around a very thin solid object for which incorrect results were obtained with previous IB methods. In order to have sufficient near-wall mesh resolution for LES and DNS computations, the present approach uses local mesh refinement. The present method has been also successfully applied to computation of the wall-pressure space–time correlation in DNS of turbulent channel flow on grids not aligned with the boundaries. When applied to a turbulent flow around an airfoil, the computed flow statistics – the mean/RMS flow field and power spectra of the wall pressure – are in good agreement with experiment.     

Nested Cartesian grid method in incompressible viscous fluid flow

In this work, the local grid refinement procedure is focused by using a nested Cartesian grid formulation. The method is developed for simulating unsteady viscous incompressible flows with complex immersed boundaries. A finite-volume formulation based on globally second-order accurate central-difference schemes is adopted here in conjunction with a two-step fractional-step procedure. The key aspects that needed to be considered in developing such a nested grid solver are proper imposition of interface conditions on the nested-block boundaries, and accurate discretization of the governing equations in cells that are with block-interface as a control-surface. The interpolation procedure adopted in the study allows systematic development of a discretization scheme that preserves global second-order spatial accuracy of the underlying solver, and as a result high efficiency/accuracy nested grid discretization method is developed. Herein the proposed nested grid method has been widely tested through effective simulation of four different classes of unsteady incompressible viscous flows, thereby demonstrating its performance in the solution of various complex flow–structure interactions. The numerical examples include a lid-driven cavity flow and Pearson vortex problems, flow past a circular cylinder symmetrically installed in a channel, flow past an elliptic cylinder at an angle of attack, and flow past two tandem circular cylinders of unequal diameters. For the numerical simulations of flows past bluff bodies an immersed boundary (IB) method has been implemented in which the solid object is represented by a distributed body force in the Navier–Stokes equations. The main advantages of the implemented immersed boundary method are that the simulations could be performed on a regular Cartesian grid and applied to multiple nested-block (Cartesian) structured grids without any difficulty. Through the numerical experiments the strength of the solver in effectively/accurately simulating various complex flows past different forms of immersed boundaries is extensively demonstrated, in which the nested Cartesian grid method was suitably combined together with the fractional-step algorithm to speed up the solution procedure.     

Boundary data immersion method for Cartesian-grid simulations of fluid-body interaction problems

A new robust and accurate Cartesian-grid treatment for the immersion of solid bodies within a fluid with general boundary conditions is described. The new approach, the Boundary Data Immersion Method (BDIM), is derived based on a general integration kernel formulation which allows the field equations of each domain and the interfacial conditions to be combined analytically. The resulting governing equation for the complete domain preserves the behavior of the original system in an efficient Cartesian-grid method, including stable and accurate pressure values on the solid boundary. The kernel formulation allows a detailed analysis of the method, and it is demonstrated that BDIM is consistent, obtains second-order convergence relative to the kernel width, and is robust with respect to the grid and boundary alignment. Formulation for no-slip and free slip boundary conditions are derived and numerical results are obtained for the flow past a cylinder and the impact of blunt bodies through a free surface. The BDIM predictions are compared to analytic, experimental and previous numerical results confirming the properties, efficiency and efficacy of this new boundary treatment for Cartesian grid methods.     

A Sharp-Interface Immersed Boundary Method with Improved Mass Conservation and Reduced Spurious Pressure Oscillations

A method for reducing the spurious pressure oscillations observed when simulating moving boundary flow problems with sharp-interface immersed boundary methods (IBMs) is proposed. By first identifying the primary cause of these oscillations to be the violation of the geometric conservation law near the immersed boundary, we adopt a cut-cell based approach to strictly enforce geometric conservation. In order to limit the complexity associated with the cut-cell method, the cut-cell based discretization is limited only to the pressure Poisson and velocity correction equations in the fractional-step method and the small-cell problem tackled by introducing a virtual cell-merging technique. The method is shown to retain all the desirable properties of the original finite-difference based IBM while at the same time, reducing pressure oscillations for moving boundaries by roughly an order of magnitude.     

A Brinkman penalization method for compressible flows in complex geometries

To simulate flows around solid obstacles of complex geometries, various immersed boundary methods had been developed. Their main advantage is the efficient implementation for stationary or moving solid boundaries of arbitrary complexity on fixed non-body conformal Cartesian grids. The Brinkman penalization method was proposed for incompressible viscous flows by penalizing the momentum equations. Its main idea is to model solid obstacles as porous media with porosity, , and viscous permeability approaching zero. It has the pronounced advantages of mathematical proof of error bound, strong convergence, and ease of numerical implementation with the volume penalization technique. In this paper, it is extended to compressible flows. The straightforward extension of penalizing momentum and energy equations using Brinkman penalization with respective normalized viscous, η, and thermal, η_T, permeabilities produces unsatisfactory results, mostly due to nonphysical wave transmissions into obstacles, resulting in considerable energy and mass losses in reflected waves. The objective of this paper is to extend the Brinkman penalization technique to compressible flows based on a physically sound mathematical model for compressible flows through porous media. In addition to penalizing momentum and energy equations, the continuity equation for porous media is considered inside obstacles. In this model, the penalized porous region acts as a high impedance medium, resulting in negligible wave transmissions. The asymptotic analysis reveals that the proposed Brinkman penalization technique results in the amplitude and phase errors of order O((η)^1/2) and O((η/η_T)^1/43/4), when the boundary layer within the porous media is respectively resolved or unresolved. The proposed method is tested using 1- and 2-D benchmark problems. The results of direct numerical simulation are in excellent agreement with the analytical solutions. The numerical simulations verify the accuracy and convergence rates.     

High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy

In this paper, a finite difference code for Direct and Large Eddy Simulation (DNS/LES) of incompressible flows is presented. This code is an intermediate tool between fully spectral Navier–Stokes solvers (limited to academic geometry through Fourier or Chebyshev representation) and more versatile codes based on standard numerical schemes (typically only second-order accurate). The interest of high-order schemes is discussed in terms of implementation easiness, computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations. The equivalence rules between operations in physical and spectral spaces are efficiently used to solve the Poisson equation introduced by the projection method. It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip. Using the concept of the modified wave number, the incompressibility can be enforced up to the machine accuracy. The benefit offered by this alternative method is found to be very satisfactory, even when a formal second-order error is introduced locally by boundary conditions that are neither periodic nor symmetric. The usefulness of high-order schemes combined with an immersed boundary method (IBM) is also demonstrated despite the second-order accuracy introduced by this wall modelling strategy. In particular, the interest of a partially staggered mesh is exhibited in this specific context. Three-dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high-order schemes to reduce the computational cost for a given accuracy. The main conclusion of this paper is that finite difference schemes with quasi-spectral accuracy can be very efficient for DNS/LES of incompressible flows, while allowing flexibility for the boundary conditions and easiness in the code development. Therefore, this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.     

An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments

We have introduced a modified penalty approach into the flow-structure interaction solver that combines an immersed boundary method (IBM) and a multi-block lattice Boltzmann method (LBM) to model an incompressible flow and elastic boundaries with finite mass. The effect of the solid structure is handled by the IBM in which the stress exerted by the structure on the fluid is spread onto the collocated grid points near the boundary. The fluid motion is obtained by solving the discrete lattice Boltzmann equation. The inertial force of the thin solid structure is incorporated by connecting this structure through virtual springs to a ghost structure with the equivalent mass. This treatment ameliorates the numerical instability issue encountered in this type of problems. Thanks to the superior efficiency of the IBM and LBM, the overall method is extremely fast for a class of flow-structure interaction problems where details of flow patterns need to be resolved. Numerical examples, including those involving multiple solid bodies, are presented to verify the method and illustrate its efficiency. As an application of the present method, an elastic filament flapping in the Kármán gait and the entrainment regions near a cylinder is studied to model fish swimming in these regions. Significant drag reduction is found for the filament, and the result is consistent with the metabolic cost measured experimentally for the live fish.     

Shape oscillations of a droplet in an Oldroyd-B fluid

We present a Navier-Stokes/Oldroyd-B immersed boundary algorithm that captures the interaction of a flexible structure with a viscoelastic fluid. In particular, we study the effects of bulk viscoelasticity on freely decaying shape oscillations of an Oldroyd-B fluid droplet suspended in an Oldroyd-B matrix. Our numerical data indicate that if the fluid viscosity is low, viscoelasticity plays a modulating role in the drop shape relaxation; specifically, it increases the oscillation frequency and decreases the decay rate when the fluid relaxation time is above a critical value. In the high viscosity limit, i.e., when a Newtonian droplet is expected to return to a spherical shape with an aperiodic decay, an increase in the relaxation time eventually results in the reappearance of the oscillations. Both these results are in line with the prediction of small deformation theory for viscoelastic droplet oscillations. The algorithm was also validated by direct comparison with linear asymptotics.     

Experimental studies on the interaction of jets with the vortex wake of slender bodies

An experimental study has been conducted to examine the low-speed aerodynamic characteristics of a pair of circular jets behaviour in the vortex-wake of a blunted cone cylinderbody combination with and without a 70° leading edge sweep delta wing. The induced flow-field was examined via two-dimensional hot-wire anemometry surveys. Comparisons were made between the induced wake with and without the circular jets. The flow field over the models was visualized using smoke and oil flow visualization. The jet-pair was found to undergo severe distortion within a very short distance from the nozzle exits. The shape of the jets was shown to change considerably when placed in the vortex wake flow field of the models, which caused the jet-pair to spread and become entrained within the vortices and wake created by the models.     

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged, leads to a linear system of equations for the interface configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin’s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system’s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix–vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve.     

An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries

We present an immersed interface method for solving the incompressible steady Stokes equations involving fixed/moving interfaces and rigid boundaries (irregular domains). The fixed/moving interfaces and rigid boundaries are represented by a number of Lagrangian control points. In order to enforce the prescribed velocity at the rigid boundaries, singular forces are applied on the fluid at these boundaries. The strength of singular forces at the rigid boundary is determined by solving a small system of equations. For the deformable interfaces, the forces that the interface exerts on the fluid are calculated from the configuration (position) of the deformed interface. The jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed/moving interfaces and rigid boundaries. These forces are interpolated using cubic splines and applied to the fluid through the jump conditions. The positions of the deformable interfaces are updated implicitly using a quasi-Newton method (BFGS) within each time step. In the proposed method, the Stokes equations are discretized via the finite difference method on a staggered Cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible Stokes flows with fixed/moving interfaces on irregular domains.