Stokes Solutions Research Articles

A gas-kinetic theory based multidimensional high-order method for the compressible Navier–Stokes solutions

This paper presents a gas-kinetic theory based multidimensional high-order method for the compressible Naiver–Stokes solutions. In our previous study, a spatially and temporally dependent third-order flux scheme with the use of a third-order gas distribution function is employed. However, the third-order flux scheme is quite complicated and less robust than the second-order scheme. In order to reduce its complexity and improve its robustness, the second-order flux scheme is adopted instead in this paper, while the temporal order of method is maintained by using a two stage temporal discretization. In addition, its CPU cost is relatively lower than the previous scheme. Several test cases in two and three dimensions, containing high Mach number compressible flows and low speed high Reynolds number laminar flows, are presented to demonstrate the method capacity.

Extensional channel flow revisited: a dynamical systems perspective

Extensional self-similar flows in a channel are explored numerically for arbitrary stretching-shrinking rates of the confining parallel walls. The present analysis embraces time integrations, and continuations of steady and periodic solutions unfolded in the parameter space. Previous studies focused on the analysis of branches of steady solutions for particular stretching-shrinking rates, although recent studies focused also on the dynamical aspects of the problems. We have adopted a dynamical systems perspective, analysing the instabilities and bifurcations the base state undergoes when increasing the Reynolds number. It has been found that the base state becomes unstable for small Reynolds numbers, and a transitional region including complex dynamics takes place at intermediate Reynolds numbers, depending on the wall acceleration values. The base flow instabilities are constitutive parts of different codimension-two bifurcations that control the dynamics in parameter space. For large Reynolds numbers, the restriction to self-similarity results in simple flows with no realistic behaviour, but the flows obtained in the transition region can be a valuable tool for the understanding of the dynamics of realistic Navier-Stokes solutions.

Direct Numerical Simulation Based Analysis of RANS Predictions of a Low-Pressure Turbine Cascade

The state-of-the-art design of turbomachinery components is based on Reynolds-averaged Navier–Stokes (RANS) solutions. RANS solvers model the effects of turbulence and boundary layer transition and therefore allow for a rapid prediction of the aerodynamic behavior. The only drawback is that modeling errors are introduced to the solution. Researchers and computational fluid dynamics developers are working on reducing these errors by improved model calibrations which are based on experimental data. These experiments do not typically, however, offer detailed insight into three-dimensional flow fields and the evolution of model quantities in an actual machine. This can be achieved through a direct step-by-step comparison of model quantities between RANS and direct numerical simulation (DNS). In the present work, the experimentally obtained model correlations are recomputed based on DNS of the same turbine profile simulated by RANS. The actual local values are compared to the modeled RANS results, providing information about the source of model deficits. The focus is on the transition process on the blade suction side (SS) and on evaluating the development of turbulent flow structures in the blade's wake. It is shown that the source of disagreement between RANS and DNS can be traced back to three major deficiencies that should be the focus of further model improvements.

Extendable outflow boundary conditions for dissipative particle dynamics simulation

This paper presents a systematic study on a basic kind of outflow boundary condition for Dissipative Particle Dynamics (DPD) simulation, in which the flow near the outlet can be assumed as fully developed. The boundary conditions are expressed in Neumann-type constraints due to the zero gradients of hydrodynamic quantities on the boundary. The flow rate is included as an additional constraint to ensure the stability of a simulation. The boundary conditions are imposed through a reservoir, which is a replica of the simulation domain in the vicinity of the outlet. All the particle configurations are conserved in the reservoir. A velocity adjusting scheme is developed for the reservoir in order to control the deviation of the outflow rate and to remove the pressure discontinuity across the boundary. The effectiveness of the boundary method is examined through the simulations of a uniform flow and a laminar flow in a bifurcated tunnel. The outflow rates, as well as the velocity profile and pressure distribution, all agree well with the Navier–Stokes solution by comparison. The assumption of the fully developed outflow is validated by measuring the gradients of the flow rates. It is expected that this boundary method can be used in the simulation of flows with heterogeneous particle distribution, without having to evaluate the distribution function in advance.

Using steady flow analysis for noise predictions

Abstract Three different methods based on steady Reynolds-averaged Navier–Stokes (RANS) solutions are used to calculate the noise emitted from airfoils. Their results are compared to the ones obtained from experiments, a semi-empirical airfoil self-noise prediction code called NAFNoise developed by NREL, and large eddy simulations (LES). The three methods considered are a noise metric developed by Hosder et al. which can only predict overall sound pressure levels (OASPL) but no frequency spectra and two different statistical models developed by Doolan et al. and Remmler et al. The method by Doolan et al. employs a Green’s function solution for airfoil trailing-edge far-field noise whereas the method by Remmler et al. predicts the pressure spectrum on the airfoil surface which is then used to compute the far-field sound by means of a hybrid noise prediction. All noise predictions were made at low speed and moderate Reynolds number similar to the environment of a small unmanned aerial system and involved different NACA airfoils as well as the SD 7003 airfoil.

Remarks on the Inviscid Limit for the Navier--Stokes Equations for Uniformly Bounded Velocity Fields

We consider the vanishing viscosity limit of the Navier--Stokes equations in a half-plane, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the product of the components of the Navier--Stokes solutions are equicontinuous at $x_2=0$. A sufficient condition for this to hold is that the tangential Navier--Stokes velocity remains uniformly bounded and has a uniformly integrable tangential gradient near the boundary.

A Discontinuous Galerkin Reduced Basis Numerical Homogenization Method for Fluid Flow in Porous Media

We present a new conservative multiscale method for Stokes flow in heterogeneous porous media. The method couples a discontinuous Galerkin finite element method (DG-FEM) at the macroscopic scale for the solution of an effective Darcy equation with a Stokes solver at the pore scale to recover effective permeabilities at macroscopic quadrature points. To avoid costly computation of numerous Stokes problems throughout the macroscopic computational domain, the pore geometry is parametrized and a model order reduction algorithm is used to select representative microscopic geometries. Accurate Stokes solutions and related permeabilities are obtained for these representative geometries in an offline stage. In an online stage, the DG-FEM is computed with permeabilities recovered at the desired macroscopic quadrature points from the precomputed Stokes solutions. The multiscale method is shown to be mass conservative at the macro scale and the computational cost for the online stage is similar to the cost of solving a single-scale Darcy problem. Numerical experiments for two- and three-dimensional problems illustrate the efficiency and the performance of the proposed method.

Numerical and experimental comparison of complete three-dimensional particle tracking velocimetry algorithms for indoor airflow study

The experimental data retrieved from three-dimensional particle tracking velocimetry (3D PTV) are crucial for indoor environment engineering when designing ventilation strategies or monitoring airborne pollutants dispersion in inhabited spaces as well as in livestock compounds. The performance of the measurement algorithm, as the core part of the 3D PTV technology, has a strong influence on the final results. This study presents a method to select the better 3D PTV algorithm depending on the application targeted. This study first compared the performance of seven 3D PTV algorithms on two types of 3D laminar macro scale flows of known Navier–Stokes solutions, namely the Kovasznay and Beltrami flows. The comparison was based on the accuracy, the measuring volume coverage and the achievable trajectory length of each algorithm with respect to the particle tracking density and the total number of frames. Then, the measurement algorithms with better performances were further compared with the experimental measurement of real 3D airflows in a cubic cavity. The results suggest that different measurement algorithms have different advantages and drawbacks depending on the application targeted, and that the algorithms using the epipolar constraint as the method of spatial matching generally perform better.

Modeling self-assembly and capture phenomenon of two droplets in high aspect ratio microchannels

Abstract Droplets will self-organize into stable and periodic arrangements of crystalline morphology when moving in high aspect ratio microchannels. Theoretical analysis of the behaviors is necessary to find the assembly mechanism and for applications. Here, we do modeling study to the experimental phenomenon that two droplets with different sizes show dynamic self-assembly or capture behaviors in high aspect ratio microchannels. Three stages (disturbance, self-assembly and relative stability) are used to describe the droplets motion process and the Stokes solutions are obtained in tangent sphere coordinates. Comparisons of the forces verify the validity of the theoretical results and trajectory analysis shows the dynamic assembly or capture process of droplets. We find that an arbitrary small turbulence changes the forces acting on droplets, breaks the equilibrium, and finally, the two droplets form a stable arrangement and move together. The relative motion of droplets contributes to the assembly or capture process. This work offers theoretical explanation of the experimental results, and can be conducted for more droplets assembly studies and probable applications such as the synthesis of new particle materials and information expression.

The planar Couette flow with slip and jump boundary conditions in a microchannel

Abstract The steady state of a dilute gas enclosed within a rectangular cavity, whose upper and lower sides are in relative motion, is considered in the slip and early transition regimes. The DSMC (Direct simulation Monte Carlo) method is used to solve the Boltzmann equation for analysing a Newtonian viscous heat conducting ideal gas with the slip and jump boundary conditions (SJBC) in the vicinity of horizontal walls. The numerical results are compared with the Navier–Stokes solutions, with and without SJBC, through the velocity, temperature, and normal heat flux profiles. The parallel heat flux and shear stress are also evaluated as a function of rarefaction degree; estimated by the Knudsen number K n ${K_{n}}$ . Thus, the breakdown of the classical Navier–Stokes theory is clarified in the non-equilibrium area, so-called Knudsen layer, near the top and bottom sides.

A parabolic velocity-decomposition method for wind turbines

An economical parabolized Navier–Stokes approximation for steady incompressible flow is combined with a compatible wind turbine model to simulate wind turbine flows, both upstream of the turbine and in downstream wake regions. The inviscid parabolizing approximation is based on a Helmholtz decomposition of the secondary velocity vector and physical order-of-magnitude estimates, rather than an axial pressure gradient approximation. The wind turbine is modeled by distributed source-term forces incorporating time-averaged aerodynamic forces generated by a blade-element momentum turbine model. A solution algorithm is given whose dependent variables are streamwise velocity, streamwise vorticity, and pressure, with secondary velocity determined by two-dimensional scalar and vector potentials. In addition to laminar and turbulent boundary-layer test cases, solutions for a streamwise vortex-convection test problem are assessed by mesh refinement and comparison with Navier–Stokes solutions using the same grid. Computed results for a single turbine and a three-turbine array are presented using the NREL offshore 5-MW baseline wind turbine. These are also compared with an unsteady Reynolds-averaged Navier–Stokes solution computed with full rotor resolution. On balance, the agreement in turbine wake predictions for these test cases is very encouraging given the substantial differences in physical modeling fidelity and computer resources required.

Numerical investigation of flexible flapping wings using computational fluid dynamics/computational structural dynamics method

A fluid–structure interaction numerical simulation was performed to investigate the flow field around a flexible flapping wing using an in-house developed computational fluid dynamics/computational structural dynamics solver. The three-dimensional (3D) fluid–structure interaction of the flapping locomotion was predicted by loosely coupling preconditioned Navier–Stokes solutions and non-linear co-rotational structural solutions. The computational structural dynamic solver was specifically developed for highly flexible flapping wings by considering large geometric non-linear characteristics. The high fidelity of the developed methodology was validated by benchmark tests. Then, an analysis of flexible flapping wings was carried out with a specific focus on the unsteady aerodynamic mechanisms and effects of flexion on flexible flapping wings. Results demonstrate that the flexion will introduce different flow fields, and thus vary thrust generation and pressure distribution significantly. In the meanwhile, relationship between flapping frequency and flexion plays an important role on efficiency. Therefore, appropriate combination of frequency and flexion of flexible flapping wings provides higher efficiency. This study may give instruction for further design of flexible flapping wings.

An updated-Lagrangian damage mechanics formulation for modeling the creeping flow and fracture of ice sheets

An updated-Lagrangian formulation is developed to model the incompressible Stokes (creeping) flow and fracture of ice sheets using both explicit-integral and implicit-gradient nonlocal damage approaches. The governing equations of incompressible nonlinearly viscous Stokes flow assuming the plane strain approximation are discretized using high-order mixed finite elements over the current reference domain. The discretized nonlinear system is solved using a Picard iteration scheme, and a mesh update method is employed to obtain the updated reference domain at every time step. Fracture (crevasse) initiation and propagation is modeled using a scalar (isotropic) damage variable, and damage control or element removal strategies are implemented to avoid numerical accuracy and convergence issues arising from fully damaged finite elements near the crevasse tip. The formulation is implemented in the open-source finite element software FEniCS, and the relevant numerical algorithms are detailed. Numerical verification and benchmark studies based on manufactured nonlinear Stokes solutions and constant velocity and gravity-driven creep flow experiments are conducted to establish the viability of the formulation. We demonstrate that crevasse propagation rates obtained from nonlinear Stokes and Maxwell viscoelastic models are in good agreement over short time scales (days), so it is reasonable to neglect the elastic effects and employ the Stokes model to simulate iceberg calving. Furthermore, we demonstrate that over long time scales (months) the updated-Lagrangian Stokes formulation is more physically accurate than the total Lagrangian Maxwell viscoelastic formulation because the former accounts for the domain geometry changes. To conclude, the merit of the proposed formulation is two-fold: first, it is computationally more efficient than the Maxwell viscoelastic model; and second, it accounts for finite strain creep deformations accruing over long time scales.

On the fifth-order Stokes solution for steady water waves

This paper presents a universal fifth-order Stokes solution for steady water waves on the basis of potential theory. It uses a global perturbation parameter, considers a depth uniform current, and thus admits the flexibilities on the definition of the perturbation parameter and on the determination of the wave celerity. The universal solution can be extended to that of Chappelear (1961), confirming the correctness for the universal theory. Furthermore, a particular fifth-order solution is obtained where the wave steepness is used as the perturbation parameter. The applicable range of this solution in shallow depth is analyzed. Comparisons with the Fourier approximated results and with the experimental measurements show that the solution is fairly suited to waves with the Ursell number not exceeding 46.7.

Stability estimates for Navier-Stokes equations and application to inverse problems

In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method.

Three-dimensional fluid–structure interaction analysis of a flexible flapping wing under the simultaneous pitching and plunging motion

Recent advance in flapping-wing MAVs has led to greater attention being paid to the interaction between the structural dynamics of the wing and its aerodynamics, both of which are closely related to the performance of a flapping wing. In this paper, an improved computational framework to simulate a flapping wing is developed. This framework is established by coupling a preconditioned Navier–Stokes solution and a co-rotational beam analysis with a restrained warping degree of freedom. Validation of the present framework is performed by a comparison with examples from either earlier analyses or experiments. Further, a numerical analysis of a wing under simultaneous pitching and plunging motion is examined. The results are compared with those obtained with a wing under pure plunging motion, in order to assess the additional motion effect within a spanwise flexible wing. The comparison shows different aerodynamic characteristics induced by the flexibility of the wing, which can be beneficial.

An improved interface preserving level set method for simulating three dimensional rising bubble

A new interface preserving level set method is developed in three steps to simulate bubble rising problems. In the first step of the solution algorithm, the level set function ϕ is advected by a pure advection equation. An intermediate step is performed to obtain new level set function through an improved smoothed Heaviside function. To keep the new level set function as a distance function and to conserve mass bounded by the interface, in the final solution step a mass correction term is added to the re-initialization equation. This two-phase numerical model is developed underlying the projection method to compute the incompressible Navier–Stokes solutions in collocated grids. In the discretizations of the level set advection equation and the re-initialization equation, the fifth-order weighted essentially non-oscillatory scheme is applied to prevent numerical oscillations occurring around discontinuous interface. The performance of the proposed level set method in conserving mass is compared with conventional level set method applied to solve the single bubble rising problem and the bubble bursting problem at a free surface. Merger of two bubbles is also investigated. Numerical results show that not only the surface tension force can be accurately calculated but also the mass can be conserved excellently using the present level set method.

Impact of Weld Joint on Compressor Blade Performance

This contribution focused on the analysis of weld-joint effects resulting from the assembly of the blades and disk for the compressor blades. Experiments at the von Kármán Institute and computational fluid dynamics simulations at Cenaero were conducted to study the flowfield in a low-pressure high-speed compressor cascade with two weld-joint configurations. Experiments were performed to measure the total pressure loss coefficient for various inlet flow conditions. Reynolds-averaged Navier–Stokes simulations using the turbulence model were carried out. Comparison between the Reynolds-averaged Navier–Stokes solutions and the experiments were made, and the accuracy of the results was investigated. The impact of the weld-joint height and the upstream boundary layer on the losses was studied. In addition, the effect of the weld-joint heights on the growth of the boundary layer was analyzed and presented.

Block-structured compressible Navier–Stokes solution using the OPS high-level abstraction

abstractIn this paper, we report the development and validation of a compressible solver with shock capturing, using a domain-specific high-level abstraction framework, OPS, that is being developed at the University of Oxford. OPS uses an active library approach for block-structured meshes, capable of generating codes for a variety of parallel implementations with different parallelisation strategies. Performance results on various architectures are reported for the 1D Shu–Osher test case.

Grid-Convergence of Reynolds-Averaged Navier–Stokes Solutions for Benchmark Flows in Two Dimensions

A detailed grid-convergence study has been conducted to establish reference solutions corresponding to the one-equation linear eddy-viscosity Spalart–Allmaras turbulence model for two-dimensional turbulent flows around the NACA 0012 airfoil and a flat plate. The study involved the three widely used codes CFL3D (NASA), FUN3D (NASA), and TAU (DLR, The German Aerospace Center), as well as families of uniformly refined structured grids that differed in the grid density patterns. Solutions computed by different codes on different grid families appeared to converge to the same continuous limit but exhibited strikingly different convergence characteristics. The grid resolution in the vicinity of geometric singularities, such as a sharp trailing edge, was found to be the major factor affecting accuracy and convergence of discrete solutions; the effects of this local grid resolution were more prominent than differences in discretization schemes and/or grid elements. The results reported for these relatively simple turbulent flows demonstrated that CFL3D, FUN3D, and TAU solutions were very similar on the finest grids used in the study, but even those grids were not sufficient to conclusively establish an asymptotic convergence order.