Analytical Solution for Laminar Viscous Flow inthe Gap Between Two Parallel Rotary Disks

Aim To get the analytical for laminar viscous flow in the gap of two parallel rotating disks. Methods By estimating the order of magnitude of each term in the Navier-Stokes equations to drop small terms and achieve the required simplified differential equations, and by integrating the equations to obtain the solution for theflow between two rotary disks. Results Parameters related to the laminar viscous flow in the gap between two parallel rotary disks, such as the velocity, the pressure, the flowrate, the force, the shearing stress, the torque and the power derived. Conclusion The result provides a theoretical basis and an effective method for the designs of the devices connected with the laminar viscous flow in the gap between two parallel rotary disks.

GRIDLESS METHOD FOR UNSTEADY VISCOUS FLOWS

Gridless method is developed for unsteady viscous flows involving moving boundaries. The point distri- bution of gridless method is implemented in an isotropic or anisotropic way according to the features of viscous flows. In the area far away from the body, the traditional cloud of isotropic points is used, while in the adjacent area, the cloud of anisotropic points is distributed. In this way, the point spacing normal to the wall can be small enough for simulating the boundary layer, and meanwhile, the total number of points in the computational do- main can be controlled due to large spacing in other tangential direction through the anisotropic way. A fast mov- ing technique of clouds of points at each time-step is presented based on the attenuation law of disturbed motion for unsteady flows involving moving boundaries. In the mentioned cloud of points, a uniform weighted least- square curve fit method is utilized to discretize the spatial derivatives of the Navier-Stokes equations. The pro- posed gridless method, coupled with a dual time-stepping method and the Spalart-Allmaras turbulence model, is implemented for the Navier-Stokes equations. The computational results of unsteady viscous flows around a NLR7301 airfoil with an oscillating flap and a pitching NACA0012 airfoil are presented in a good agreement with the available experimental data.

Numerical Simulation of Viscous Flow Around a Rolling Cylinder with Ship-Like Section

Two-dimensional unsteady incompressible viscous flow around a rolling cylinder with ship-like section is numerically simulated by employing the computational scheme previously developed by the authors, in which the continuity and momentum equations are satisfied simultaneously at each time step for oscillating flow. The numerical results show that the motion of vortices around a rolling ship hull is cyclical. It is found that the location of the vortices is very similar to the existing experimental result. Using these simulation results, we can calculate the roll damping of ships including viscous effects.

Numerical computation and analysis of unsteady viscous flow around autonomous underwater vehicle with propellers based on sliding mesh

The flexible transmission shaft and wheel propeller are combined as the kinetic source equipment, which realizes the nmlti-motion modes of the autonomous underwater vehicle （AUV） such as vectored thruster and wheeled movement. In order to study the interactional principle between the hull and the wheel propellers while the AUV navigating in water, the computational fluid dynamics （CFD） method is used to simulate numerically the unsteady viscous flow around AUV with propellers by using the Reynolds-averaged Navier-Stokes （RANS） equations, shear-stress transport （SST） k-w model and pressure with splitting of operators （PISO） algorithm based on sliding mesh. The hydrodynamic parameters of AUV with propellers such as resistance, pressure and velocity are got, which reflect well the real ambient flow field of AUV with propellers. Then, the semi-implicit method for pressure-linked equations （SIMPLE） algorithm is used to compute the steady viscous flow field of AUV hull and propellers, respectively. The computational results agree well with the experimental data, which shows that the numerical method has good accuracy in the prediction of hydrodynamic performance. The interaction between AUV hull and wheel propellers is predicted qualitatively and quantitatively by comparing the hydrodynamic parameters such as resistance, pressure and velocity with those from integral computation and partial computation of the viscous flow around AUV with propellers, which provides an effective reference to the shady on noise and vibration of AUV hull and propellers in real environment. It also provides technical support for the design of new AUVs.

A CFD Based Investigation of the Unsteady Hydrodynamic Coefficients of 3-D Fins in Viscous Flow

The motion of the fins and control surfaces of underwater vehicles in a fluid is an interesting and challenging research subject.Typically the effect of fin oscillations on the fluid flow around such a body is highly unsteady, generating vortices and requiring detailed analysis of fluid-structure interactions.An understanding of the complexities of such flows is of interest to engineers developing vehicles capable of high dynamic performance in their propulsion and maneuvering.In the present study, a CFD based RANS simulation of a 3-D fin body moving in a viscous fluid was developed.It investigated hydrodynamic performance by evaluating the hydrodynamic coefficients (lift, drag and moment) at two different oscillating frequencies.A parametric analysis of the factors that affect the hydrodynamic performance of the fin body was done, along with a comparison of results from experiments.The results of the simulation were found in close agreement with experimental results and this validated the simulation as an effective tool for evaluation of the unsteady hydrodynamic coefficients of 3-D fins.This work can be further be used for analysis of the stability and maneuverability of fin actuated underwater vehicles.

Similarity solutions to viscous flow and heat transfer of nanofluid over nonlinearly stretching sheet

The boundary-layer flow and heat transfer in a viscous fluid containing metallic nanoparticles over a nonlinear stretching sheet are analyzed. The stretching velocity is assumed to vary as a power function of the distance from the origin. The governing partial differential equation and auxiliary conditions are reduced to coupled nonlinear ordinary differential equations with the appropriate corresponding auxiliary conditions. The resulting nonlinear ordinary differential equations （ODEs） are solved numerically. The effects of various relevant parameters, namely, the Eckert number Ec, the solid volume fraction of the nanoparticles ~, and the nonlinear stretching parameter n are discussed. The comparison with published results is also presented. Different types of nanoparticles are studied. It is shown that the behavior of the fluid flow changes with the change of the nanoparticles type.

AN EFFICIENT ALGORITHM FOR HYPERSONIC VISCOUS FLOWS

The CSCM-S algorithm proposed by Lombard et al.is a very attractive tool for solving multidimensional Euler and Navier-Stokes equations.However,it is not economical due to the use of global sweeps in the whole computational domain.In this paper we suggest a modified strategy,which combines a single-marching technique for supersonic dominated region with a multi-sweep procedure for pure subsonic and complex flowfield.The new algorithm may save significantly CPU time and is more suitable for engineering applications.

An analysis of the incompressible viscous flows problem by meshless method

Generally the incompressible viscous flow problem is described by the Navier-Stokes equation. Based on the weighted residual method the discrete formulation of element-free Galerkin is inferred in this paper. By the step-bystep computation in the field of time, and adopting the least-square estimation of the-same-order shift, this paper has calculated both velocity and pressure from the decoupling independent equations. Each time fraction Newton-Raphson iterative method is applied for the velocity and pressure. Finally, this paper puts the method into practice of the shear-drive cavity flow, verifying the validity, high accuracy and stability.

HYBRID CARTESIAN GRID/GRIDLESS METHOD FOR CALCULATING VISCOUS FLOWS OVER MULTI-ELEMENT AIRFOILS

A hybrid Cartesian grid/gridless method is developed for calculating viscous flows over multi-element airfoils.The method adopts an unstructured Cartesian grid to cover most areas of the computational domain and leaves only small region adjacent to the aerodynamic bodies to be filled with the cloud of points used in the gridless methods,which results in a better combination of the computational efficiency of the Cartesian grid and the flexibility of the gridless method in handling complex geometries.The clouds of points in the local gridless region are implemented in an anisotropic way according to the features of the thin boundary layer of the viscous flows over the airfoils,and the clouds of points at the vicinity of the interface between the grid and the gridless regions are also controlled by using an adaptive refinement technique during the generation of the unstructured Cartesian grid.An implementation of the resulting hybrid method is presented for solving two-dimensional compressible Navier-Stokes(NS)equations.The simulations of the viscous flows over a RAE2822airfoil or a two-element airfoil are successfully carried out,and the obtained results agree well with the available experimental data.

Numerical simulations of viscous flow around the obliquely towed KVLCC2M model in deep and shallow water

By solving the unsteady Reynolds averaged Navier-Stokes （RANS） equations in combination with the k- ε SST turbulence model, the unsteady viscous flow around the obliquely towed tanker KVLCC2M model in both deep and shallow waters is simulated and the hydrodynamic forces, the surface pressure distribution, and the wake field are calculated. The overset grid technology is used to avoid the grid distortion in large drift angle cases. The effects of the free surface are taken into account. At the first stage, the deep water cases with five oblique angles are designed as the benchmark test cases. The predicted wake field, the surface pressure distribution and the hydrodynamic forces acting on the hull agree well with the corresponding experimental data, implying the capability of the present method in the prediction of the viscous flow around the tanker drifting in shallow water. A set of systematic computations with varying water depths and drift angles are then carried out to study the viscous flow around the model drifting in shallow water. The forces and moments, as well as the surface pressure distribution are predicted and analyzed. The most significant changes such as the increased stagnation pressure in the bow, the acceleration of the flow along the ship＇s sides and in the gap between ship and seabed, the lower hull pressure and finally, the stronger vortices along the bilges and weaker vortices with larger diameters in the wake are noticed.

AN ARTIFICIAL BOUNDARY CONDITION FOR THE INCOMPRESSIBLE VISCOUS FLOWS IN A NO-SLIP CHANNEL

The numerical simulation of the steady incompressible viscous flows in a no-slip channel is considered. A discrete artificial boundary condition on a given segmental artificial boundary is designed by the method of lines. Then the original problem is reduced to a boundary value problem of Navier-Stokes equations on a bounded domain. The numerical examples show that this artificial boundary condition is very effective and more accurate than Dirichlet and Neumann boundary conditions used in engineering literature.

BLOCK ITERATIVE METHODS FOR LINEAR ALGEBRAIC EQUATION AND DOMAIN DECOMPOSITION METHOD FOR INCOMPRESSIBLE VISCOUS FLOW

It was proved numerically that the Domain Decomposition Method CDDM) with one layer overlapping grids is identical to the block iterative method of linear algebra equations. The results obtained by using DDM could be in reasonable agreement with the results of full-domain simulation. With the three dimensional solver developed by the authors, the flow field in a pipe was simulated by using the full-domain DDM with one layer overlapping grids and with patched grids respectively. Both of the two cases led to the convergent solution. Further research shows the superiority of the DDM with one layer overlapping grids to the DDM with patched grids. A comparison between the numerical results obtained by the authors and the experimental results given by Enayet[3] shows that the numerical results are reasonable.

VISCOUS-INVISCID INTERACTING FLOW THEORY

In this paper a viscous-inviscid interacting flow theory(IFT)is developed for an incompressible, two-dimensional laminar flow.IFT's main points are as follows.(1)By introducing a concept of interaction lay- er where the normal momentum exchange is dominating,a new three layer structure is established.(2)Through the conventional manipulations and by introducing an interaction model,both the streamwise and normal length scales are proved to be functions of a single parameter m,which is related to the streamwise pressure gradient and Reynolds number.(3)The approximate equations governing the flow of each layer as well as the whole interaction flow are derived.The present IFT is applicable to both attached and attached-separation bubble-reattached flows, The classical boundary layer theory and Triple-deck theory are shown to be two special cases of the present theory under m=0 and 1/4,respectively.Furthermore IFT provides new distinctions of both the normal and streamwise length scales for flow-field numerical computation and also gives a new approach to developing the simpli- fied Navier-Stokes(SNS)equations.

Coupling potential and viscous flow models with domain decomposition for wave propagations

Either potential flow or viscous flow based model may be flawed for numerical wave simulations.The two-way coupling of potential and viscous flow models with the domain decomposition utilizing respective strengths has been a trending research topic.In contrast to existing literatures in which closed source potential models were used,the widely used open source OceanWave3D,OpenFOAM-v2012 are used in the present research.An innovative overlapping two-way coupling strategy is developed utilizing the ghost points in OceanWave3D.To guarantee computational stability,a relaxation zone used both for outlet damping and data transfer is built over the overlapping region in OceanWave3D.The free surface elevation in the relaxation zone is directly probed in OpenFOAM while the velocity potential is indirectly built upon its temporal variation which is calculated by the free surface boundary condition using the probed velocity.Strong coupling is achieved based on the fourth-order Runge-Kutta(RK)algorithm.Both two-and three-dimensional tests including linear,nonlinear,irregular,and multi-directional irregular waves,are conducted.The effectiveness of the coupling procedure in bidirectional data transfer is fully demonstrated,and the model is validated to be accurate and efficient,thus providing a competitive alternative for ocean wave simulations.

An implicit algorithm based on iterative modified approximate factorization method coupling with characteristic boundary conditions for solving subsonic viscous flows

An algorithm composed of an iterative modified approximate factorization(MAF(k)) method with Navier-Stokes characteristic boundary conditions(NSCBC) is proposed for solving subsonic viscous flows.A transformation on the matrix equation in MAF(k) is made in order to impose the implicit boundary conditions properly.To be in consistent with the implicit solver for the interior domain,an implicit scheme for NSCBC is formulated.The performance of the developed algorithm is investigated using spatially evolving zero pressure gradient boundary layer over a flat plate and a wall jet mixing with a cross flow over a flat plate with a square hole as the test cases.The numerical results are compared to the existing experimental datasets and a number of general correlations,together with other available numerical solutions,which demonstrate that the developed algorithm possesses promising capacity for simulating the subsonic viscous flows with large CFL number.

Hybrid grid generation for viscous flow simulations in complex geometries

In this paper,we present a hybrid grid generation approach for viscous flow simulations by marching a surface triangulation on viscous walls along certain directions.Focuses are on the computing strategies used to determine the marching directions and distances since these strategies determine the quality of the resulting elements and the reliability of the meshing procedure to a large extent.With respect to marching directions,three strategies featured with different levels of efficiencies and robustness performance are combined to compute the initial normals at front nodes to balance the trade-off between efficiency and robustness.A novel weighted strategy is used in the normal smoothing scheme,which evidently reduces the possibility of early stop of front generation at complex corners.With respect to marching distances,the distance settings at concave and/or convex corners are locally adjusted to smooth the front shape at first;a further adjustment is then conducted for front nodes in the neighbourhood of gaps between opposite viscous boundaries.These efforts,plus other special treatments such as multi-normal generation and fast detection of local/global intersection,as a whole enable the setup of a hybrid mesher that could generate qualitied viscous grids for geometries with industry-level complexities.

A three-dimensional gas-kinetic flux solver for simulation of viscous flows with explicit formulations of conservative variables and numerical flux

A truly three-dimensional(3D)gas-kinetic flux solver for simulation of incompressible and compressible viscous flows is presented in this work.By local reconstruction of continuous Boltzmann equation,the inviscid and viscous fluxes across the cell interface are evaluated simultaneously in the solver.Different from conventional gaskinetic scheme,in the present work,the distribution function at cell interface is computed in a straightforward way.As an extension of our previous work(Sun et al.,Journal of Computational Physics,300(2015)492–519),the non-equilibrium distribution function is calculated by the difference of equilibrium distribution functions between the cell interface and its surrounding points.As a result,the distribution function at cell interface can be simply calculated and the formulations for computing the conservative flow variables and fluxes can be given explicitly.To validate the proposed flux solver,several incompressible and compressible viscous flows are simulated.Numerical results show that the current scheme can provide accurate numerical results for three-dimensional incompressible and compressible viscous flows.

Computation of Viscous Uniform and Shear Flow over A Circular Cylinder by A Finite Element Method

The incompressible viscous uniform and shear flow past a circular cylinder is studied. The two-dimensional Navier-Stokes equations are solved by a finite element method. The governing equations are discretized by a weighted residual method in space. The stable three-step scheme is applied to the momentum equations in the time integration. The numerical model is firstly applied to the computation of the lid-driven cavity flow for its validation. The computed results agree well with the measured data and other numerical results. Then, it is used to simulate the viscous uniform and shear flow over a circular cylinder for Reynolds numbers from 100 to 1000. The transient time interval before the vortex shedding occurs is shortened considerably by introduction of artificial perturbation. The computed Strouhal number, drag and lift coefficients agree well with the experimental data. The computation shows that the finite element model can be successfully applied to the viscous flow problem.

PARALLEL COMPUTATION OF SUPER-SONIC BLUNT BODY VISCOUS FLOWFIELDS IN PVM

Supersonic viscous flows past blunt bodies is calculated with TVD difference scheme and implicit Lower Upper Symmetric Gauss Seidel (LU SGS) method, and parallel programming designing software platform PVM is used based on message passing to distribute a large task according to some patching strategies to a large number of processors in the network. These processors accomplish this large task together. The marked improvement of computational efficiency in networks, especially in MPP system, demonstrates the potential vitality of CFD in engineering design.

NUMERICAL STUDY OF TOPOLOGICAL STRUCTURE OF 3-D TRANSONIC VISCOUS FLOW FIELD (TVFF) INSIDE TURBINE CASCADE

The major purpose of this paper is to numerically study the complex structure of vortex system occurring within transonic turbine cascade. The transonic viscous flow in turbine cascade is simulated by solving full 3D Reynolds average N S equations, and then detailed analyses of vortex system structure are presented. Under guidance of topology, the wall limiting streamlines are employed to reveal the flow structure near the wall, and an analysis of space streamlines and cross section streamlines is given for the investigation of flow structure in the flow field. Through the analysis, the formation and evolution of the vortex system and the whole process of separation occurring within this turbine cascade are revealed