Incompressible Navier-Stokes Research Articles

A compact RBF-FD based meshless method for the incompressible Navier—Stokes equations

Meshless methods for solving fluid and fluid-structure problems have become a promising alternative to the finite volume and finite element methods. In this paper, a mesh-free computational method based on radial basis functions in a finite difference mode (RBF-FD) has been developed for the incompressible Navier—Stokes (NS) equations in stream function vorticity form. This compact RBF-FD formulation generates sparse coefficient matrices, and hence advancing solutions will in time be of comparatively lower cost. The spatial discretization of the incompressible NS equations is done using the RBF-FD method and the temporal discretization is achieved by explicit Euler time-stepping and the Crank—Nicholson method. A novel ghost node strategy is used to incorporate the no-slip boundary conditions. The performance of the RBF-FD scheme with the ghost node strategy is validated against a variety of benchmark problems, including a model fluid—structure interaction problem, and is found to be in a good agreement with the existing results. In addition, a higher-order RBF-FD scheme (which uses ideas from Hermite interpolation) is then proposed for solving the NS equations.

A solution to parabolic system with the fractional Laplacian

The existence of a solution to the parabolic system with the fractional Laplacian (−Δ) α/2, α > 0 is proven, this solution decays at different rates along different time sequences going to infinity. As an application, the existence of a solution to the generalized Navier-Stokes equations is proven, which decays at different rates along different time sequences going to infinity. The generalized Navier-Stokes equations are the equations resulting from replacing −Δ in the Navier-Stokes equations by (−Δ) m , m > 0. At last, a similar result for 3-D incompressible anisotropic Navier-Stokes system is obtained.

Thermofluid Analysis of Ultra Low Power Hotplates for a MOX Gas Sensing Device

This work presents a full three-dimensional finite-element multiphysics simulation of the conjugate heat transfer for a gas sensing device composed by a two-element array of ultra low power (ULP) metal oxide semiconductor (MOX) sensors operated in a miniaturized sampling chamber. The heat equation in a solid, the Poisson equation for the electric potential and the incompressible Navier-Stokes and energy equations for a fluid have been solved in a coupled manner. Validation of the simulation results has been performed comparing the simulated power dissipated by the array with a set of experimental data under different operating conditions. A maximum relative error of less than 7% between the simulations and the experiments has been obtained without application of any fitting strategy on the physical properties. A negligible effect on the power dissipated by the sensor, in presence of volumetric fluxes in the sampling chamber, has been observed both numerically and experimentally. Finally, a real operational condition has been simulated and examined.

Stable Fourth-Order Stream-Function Methods for Incompressible Flows with Boundaries

Fourth-order stream-function methods are proposed for the time dependent, incompressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the noslip condition. Formal error analysis is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically.

Large Friction Limit and the Inviscid Limit of 2D Navier–Stokes Equations Under Navier Friction Condition

We show that as the friction coefficient tends to infinity fast enough, the Navier friction condition recovers the no-slip boundary condition for the 2D incompressible Navier–Stokes (NS) equations in a smooth bounded domain. We also show that as the viscosity and the (Navier) friction coefficient tend to zero and infinity, respectively, satisfying certain conditions, the incompressible NS equations tend to the incompressible Euler equations in $L^2$ for initial data in $H^2$. We give explicitly the rates of convergence in $L^2$ of these two limits in terms of the viscosity and the friction coefficient.

A collocated, iterative fractional‐step method for incompressible large eddy simulation

AbstractFractional‐step methods are commonly used for the time‐accurate solution of incompressible Navier–Stokes (NS) equations. In this paper, a popular fractional‐step method that uses pressure corrections in the projection step and its iterative variants are investigated using block‐matrix analysis and an improved algorithm with reduced computational cost is developed. Since the governing equations for large eddy simulation (LES) using linear eddy‐viscosity‐based sub‐grid models are similar in form to the incompressible NS equations, the improved algorithm is implemented in a parallel LES solver. A collocated grid layout is preferred for ease of extension to curvilinear grids. The analyzed fractional‐step methods are viewed as an iterative approximation to a temporally second‐order discretization. At each iteration, a linear system that has an easier block‐LU decomposition compared with the original system is inverted. In order to improve the numerical efficiency and parallel performance, modified ADI sub‐iterations are used in the velocity step of each iteration. Block‐matrix analysis is first used to determine the number of iterations required to reduce the iterative error to the discretization error of$O\left(\Delta t^2\right)$. Next, the computational cost is reduced through the use of a reduced stencil for the pressure Poisson equation (PPE). Energy‐conserving, spatially fourth‐order discretizations result in a 7‐point stencil in each direction for the PPE. A smaller 5‐point stencil is achieved by using a second‐order spatial discretization for the pressure gradient operator correcting the volume fluxes. This is shown not to reduce the spatial accuracy of the scheme, and a fourth‐order continuity equation is still satisfied to machine precision. The above results are verified in three flow problems including LES of a temporal mixing layer. Copyright © 2008 John Wiley & Sons, Ltd.

Analysis of pulsatile blood flow in constricted bifurcated arteries with vorticity-stream function approach

This theoretical investigation deals with an analysis of pulsatile blood flow in a model bifurcated artery having a stenosis in the parent arterial lumen. The geometry of the bifurcated arterial segment with an implanted stenosis in the parent duct is given an appropriate mathematical shape with the introduction of suitable curvature at the lateral junction and the flow divider. The vascular wall deformability is duly accounted for although the development of atherosclerosis in the arteries reduces its elastic property to some extent. The streaming blood contained in the bifurcated artery is treated to be Newtonian. The flow dynamic analysis applies two-dimensional unsteady incompressible nonlinear Navier-Stokes equations rewritten in the vorticity-stream function formulation. Following a radial coordinate transformation, these equations are solved numerically by a finite difference scheme with the approximate choice of the inlet and boundary conditions in concert with the biophysical point of view. The final numerical results are highlighted at the end of the paper through the exhibition of the wall shear stress and several time-variant patterns of streamlines and vorticity contours of the flow phenomena, which are highly influenced by the severity of the stenosis and the angle of bifurcation. The applicability of the present model is thus established.

A modified conservation principles theory leading to an optimal Galerkin CFD algorithm

AbstractA modified conservation principles theory in one, then multi‐dimensions, admits the prediction of an optimally accurate algorithm construction for the unsteady incompressible Navier–Stokes (INS) equations. Via a time Taylor series (TS) operation, followed by a pseudo‐limit process, the theory generates a modified, but still analytical, INS system parameterized by a set of coefficients constrained only by a convexity requirement. A spatially discretized finite element implementation of a Galerkin weak statement on this modified INS system, termed the ‘Taylor weak statement (TWS), ’ generates a parameterized CFD algorithm for analysis. TWS algorithm phase velocity and amplification factor error functions are derived for linear and bilinear basis implementations assembled at the generic node. A subsequent TS expansion in wave number space admits analytical identification of parameter set options affecting lowest order error terms. The results of definitive verification‐ and validation‐class computational experiments for a range of published CFD algorithms belonging to the TWS class, reported herein, clearly confirm theoretical prediction of the optimal TWS algorithm for INS thermal/fluid transport applications. Copyright © 2007 John Wiley & Sons, Ltd.

Numerical study on the improvement of resistance performance for fast-ferry with transom appendage

In recent, stern wedges and stern flaps are installed for the improvement of propulsion and resistance performance of fast-ferry. For example, U.S. Navy has achieved the development of stern wedges and stern flaps for destroyer to enhance powering performance. It is generally known that stern wave systems as well as bow wave systems play an important role in the wave making resistance performance for fast-ferry. The bow diverging wave system has been usually simulated by an interface tracking method (ITM). However, it is difficult to apply the ITM to the numerical simulation of the stern wave and spray phenomenon because of over-turning wave and wave-breaking. Therefore, to solve this problem an interface capturing method (ICM) is introduced. In the present study, a numerical method with the ICM is developed to evaluate the resistance performance of fast-ferry. Incompressible Navier-Stokes and continuity equations are employed in the present study and the equations are discretized by Finite Difference Method in the general curvilinear coordinate system. CIP (Constrained Interpolated Profile) method is used for the discretization of convection terms, respectively. The free surface location is determined by level set method. In order to validate the numerical method, numerical simulations for Wigley hull are performed and their results are compared with experimental results. Several numerical simulations of ship waves for fast-ferry are performed to find advantages of appendage installation. Through those simulations, the computed results, such as wave profile and resistance coefficient, are compared with the measured results which are achieved from Samsung Ship Model Basin (SSMB). The effects of transom appendage on the resistance performance are discussed with the computed results in this study.

Investigation of coherent structures in rotating Rayleigh-Benard convection

The current work involves identification of coherent structures in rotating turbulent Rayleigh-Benard convection (RBC) in a moderately large aspect-ratio (8:8:1) rectangular enclosure. The enclosure is rotating about a vertical axis passing through its center of gravity. The incompressible Navier-Stokes and energy equations are solved in a rotating frame of reference and the resulting velocity and thermal fields are analyzed to educe coherent structures. The flow structures have been investigated at different nondimensional rotation rates ranging from ω=0 to 104 for a fixed Rayleigh number Ra=107 and Prandtl number Pr=0.01, keeping the Raw∕Ta ratio constant at 10−3 in order to stringently maintain Boussinesq approximation as well as to attain experimentally realizable rotational Rayleigh numbers. Coherent structures in the flow domain have been sought using different identification techniques, namely large eddy simulation (LES) decomposition using top-hat filter, proper orthogonal decomposition (POD) considering larger energy modes, second invariant (Q) of the velocity gradient tensor, and regions of negative λ2, the second largest eigenvalue of the tensor SikSkj+ΩikΩkj. It has been found that the coherent structures educed using POD or LES decomposition at low to moderate rotation (ω=10 to 103) show the formation of two to three large-scale rolls aligned along both horizontal directions. At higher-rotation rates corresponding to ω=104, there is a breakup of large-scale structures into multiple small-scale rolls having random spatial orientation. The thermal structures educed using both POD and large-scale LES decomposition at zero rotation show randomly rising and descending plumes that at Ω=10 coalesce to form a large cylindrical thermal plume in the core of the cavity. Further increase of rotation leads again to breakup of the cylindrical plume into multiple random plumes. Isosurfaces of Q and λ2 reveal elongated tubular roll-like structures mainly concentrated near the side-wall regions, though at higher rotation-rate (ω=104) the density of tubular structures increases significantly. There is a strong similarity of results obtained by large-scale LES decomposition and results obtained using mean plus first mode of POD decomposition. The structures educed using Q and λ2 are different in shape compared to LES or POD educed structures.

An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations

Discontinuous Galerkin (DG) methods have proved to be well suited for the construction of robust high-order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems. Their application to the incompressible Navier–Stokes (INS) equations has also been recently considered, although the subject is far from being fully explored. In this work, we propose a new approach for the DG numerical solution of the INS equations written in conservation form. The inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the (exact) solution of the Riemann problem associated with a local artificial compressibility perturbation of the equations. Unlike in most of the existing methods, artificial compressibility is here introduced only at the interface flux level, therefore resulting in a consistent discretization of the INS equations irrespectively of the amount of artificial compressibility introduced. The discretization of the viscous term follows the well-established DG scheme named BR2. The performance and the accuracy of the method are demonstrated by computing the Kovasznay flow and the two-dimensional lid-driven cavity flow for a wide range of Reynolds numbers and for various degrees of polynomial approximation.

Vertically integrated numerical model to simulate breach formation by flow overtopping earth-filled dams: hydraulics

The simulation of earth-filled dam breach formation and the flood waves that are generated mainly involves three physical aspects: hydraulics, sediment transport, and slope stability. Herein, a numerical model capable of simulating the hydraulics of a flow overtopping an earth-filled dam is presented. The model is of the shallow-water type, obtained by the vertical integration of the incompressible free-surface Navier-Stokes equations, but without the usual assumptions of hydrostatic pressure and small slope. Different assumptions concerning velocity, pressure, and shear stress profiles are made in order to enhance the shallow water equations by including the geometrical effects (steep slopes), the dynamic pressure effects, and the shear-stress variation effects. Another important feature of the model is the capacity to capture flow regime changes (from sub-critical to super-critical and vice versa through a hydraulic jump) in a continuous way. A stabilized finite-element method is used to solve these equations and control the model. Results obtained for the simulation of overtopping flow are presented.

NUMERICAL SIMULATION OF VENTRICULAR CANNULATION

The interface between the Left Ventricle (LV) and the apically-placed cannula is a critical link in the hemodynamic performance of Left Ventricular Assist Devices (LVADs). Yet, this interface has not received much formal attention; the hemodynamics and anatomic interaction within the LV apex is virtually unknown. Due to the high likelihood of mural thrombosis and endocardial trauma - especially with turbodynamic circulatory support - we undertook a numerical study to elucidate the associated hemodynamics. The flow field is solved numerically using commercially available Computational Fluid Dynamics (CFD) software. The blood is treated as a Newtonian fluid, governed by the incompressible Navier-Stokes (N-S) equations. The cannula tip was considered to be rigid, and the ventricular walls were compliant, yet with prescribed motion. Flow features of particular interests are recirculation and the stagnation regions in crevices between the cannula tip and endocardial surface. These results are compared to previously-acquired flow-visualization results (Antaki et al.), and entail comparative evaluation of various ventricular geometries with different cannula tips. Our subsequent studies will extend these simulations to consider coupled fluid-structure interaction to thereby accurately simulate the phenomena associated with suction.

A diffuse-interface model for axisymmetric immiscible two-phase flow

A diffuse-interface model is considered for solving axisymmetric immiscible two-phase flow with surface tension. The Navier–Stokes (NS) equations are modified by the addition of a continuum forcing. The interface between the two fluids is considered as the half level set of a mass concentration c, which is governed by the Cahn–Hilliard (CH) equation––a fourth order, degenerate, nonlinear parabolic diffusion equation. In this work, we develop a nonlinear multigrid method to solve the CH equation with degenerate mobility and couple this to a projection method for the incompressible NS equations. The diffuse-interface method can deal with topological transitions such as breakup and coalescence smoothly without ad hoc `cut and connect' or other artificial procedures. We present results for Rayleigh's capillary instability up to forming satellite drops. The results agree well with the linear stability theory.

Computational Evaluation of the Periodic Performance of a NACA 0012 Fitted With a Gurney Flap

A detailed computational investigation into the periodic two-dimensional performance of a NACA 0012 section fitted with 2 and 4 percent h/c Gurney flaps operating at a Reynolds number of 0.85×106 is presented. The aim of the work was to determine the suitability of the incompressible Reynolds-averaged Navier-Stokes (RANS) formulation in modeling the vortex shedding experienced by lifting sections with blunt, sharp edged features. In particular, whether under-converged steady state calculations could be used for section design performance evaluation in place of the computationally intensive time accurate flow simulations. Steady, periodic, and time-averaged two-dimensional lift and drag coefficients, as well as vortex shedding frequency, were predicted and compared with the available experimental data. Reasonable agreement was found, once sufficiently fine grids had been generated, and the correct time step determined for the time accurate simulations.

Computational Design of Vibration Pumping Device for Artificial Heart

To develop the prototype vibration pumping device for an artificial heart (Hashimoto et al., 1994, ASME J. Fluids Eng., 116, pp. 741–745), the flow patterns in the casing were analyzed experimentally and numerically from the viewpoint of biomechanical engineering. Considering not only the mechanical performance of the pump, but also the hemolysis, was very important to design the artificial heart. In the present study, the curvilinear coordinate transformation technique and the finite difference technique were used to numerically solve the unsteady, incompressible, and axisymmetric Navier-Stokes equations for the flow field in the various casing configurations of the vibration pumping device. The validity of numerical analysis was confirmed by comparison with the experimental data obtained by the flow visualization technique. Furthermore, the strong dependence of the hemolysis on the flow patterns in the casing was recognized. In particular, the relationship between the vorticity field in the casing and the hemolysis was elucidated. The results obtained here would provide the useful suggestions for future research and the basic design concept of vibration pumping device for the left ventricular assist device.

Meshless method for the stationary incompressible Navier-Stokes equations

Mathematical analysis is achieved on a meshless method for the stationary incompressible Stokes and Navier-Stokes equations. In particular, the Moving Least Square Reproducing Kernel (MLSRK) method is employed. The existence of discrete solution and its error estimate are obtained. As a numerical example for convergence analysis, we compute the numerical solutions for these equations to compare with exact solutions. Also we solve the driven cavity flow numerically as a test problem.

Numerical solutions of incompressible Euler and Navier-Stokes equations by efficient discrete singular convolution method

An efficient discrete singular convolution (DSC) method is introduced to the numerical solutions of incompressible Euler and Navier-Stokes equations with periodic boundary conditions. Two numerical tests of two-dimensional Navier-Stokes equations with periodic boundary conditions and Euler equations for doubly periodic shear layer flows are carried out by using the DSC method for spatial derivatives and fourth-order Runge-Kutta method for time advancement, respectively. The computational results show that the DSC method is efficient and robust for solving the problems of incompressible flows, and has the potential of being extended to numerically solve much broader problems in fluid dynamics.

Numerical study of unsteady stenosis flow: parametric evaluation of power-law model

Currently the best indicator for surgical treatment of arteriosclerosis is the degree of stenosis. Although X-ray angiography is currently the standard, cost and morbidity are distinct disadvantages. By modelling stenosis and studying its biofluid mechanics, one can apply its results in the field of arterial disease research. This formed the motivation for this work. A non-Newtonian (power law) incompressible Navier-Stokes (N- S) solver was developed using the method of operator splitting and artificial compressibility. The vehicle used is the computational fluid dynamics (CFD) numerical library FASTFLO. The power-law model developed is then used to do a parametric study of the effect of 'n' on blood flow mechanics where 'n' is the power index that determines the haematocrit of blood. A pulsatile pressure wave over a cardiac cycle of a second was used to simulate transient flow over a hypothetical twodimensional stenotic geometry. By comparing the different velocity pressure, wall shear stress and viscosity profiles, it has been found when 'n' increases, the vortex formation and peak wall shear stress decreases (magnitudes of < 1.5 Pa). Since the formation of vortices and lowoscillatory wall shear stress on the stenotic wall is detrimental to the well-being of the arterial tract, it can therefore be inferred that there might be a relationship between the diseased state of blood (power law) and early genesis of atherosclerosis. However, the conclusion of this paper marks the advent of new research directions in this field of study.

Numerical Computation of Tip Vortex Flow Generated by a Marine Propeller

The uniform flow past a rotating marine propeller was studied using incompressible Reynolds-averaged Navier-Stokes computations with the Baldwin-Barth turbulence model. Extensive comparison with the experimental data was made to validate the numerical results. The general characteristics of the propeller flow were well predicted. The current numerical method, however, produced an overly diffusive and dissipative tip vortex core. Modification of the Baldwin-Barth model to better predict the Reynolds stress measurements also improved the prediction of the mean velocity field. A modified tip geometry was also tested to show that an appropriate cross section design can delay cavitation inception in the tip vortex without reducing the propeller performance.