Thin-layer Navier-Stokes Equations Research Articles

Numerical simulation of axisymmetric supersonic viscous flow over a blunt cone with a diagonal fourth-order finite difference method

The numerical solution of steady viscous supersonic axisymmetric flowfield modelled by thin-layer Navier–Stokes equations is computed over a blunt cone with the shock-fitting method and the diagonal fourth-order central difference scheme implemented. Owing to the presence of high-order terms of the Taylor series in the discretization of derivatives, this method has high accuracy and low numerical error (dispersion error) compared with low-order methods. The boundary-closure scheme plays an important role in the numerical stability of this method. Using a coarse grid in this method, the results of numerical solution are found to be very close to those obtained with a fine grid employing the implicit second-order (Beam–Warming) method. Higher accuracy of this method is identified relative to the second-order method when the grid is being refined. The convergence rate of this method is also higher than the second-order method. Furthermore, the convergence of the method can be adjusted to accommodate the computational hardware capabilities.

Dual-code solution procedure for efficient computing equilibrium hypersonic axisymmetric transitional/turbulent flows

An appropriate combination of the thin-layer Navier–Stokes (TLNS) and parabolized Navier–Stokes (PNS) solvers is used to accurately and efficiently compute hypersonic transitional/turbulent flowfields of perfect gas and equilibrium air around blunt-body configurations. The TLNS equations are solved in the nose region to provide the initial data plane needed for the solution of the PNS equations. Then the PNS equations are employed to efficiently compute the flowfield for the afterbody region by using a space marching technique. Both the TLNS and the PNS equations are numerically solved by using the implicit non-iterative finite-difference algorithm of Beam and Warming. A shock fitting procedure is used in both the TLNS and PNS codes to obtain accurate solution in the vicinity of the shock. For turbulent flow simulations, both the Cebeci–Smith (CS) and the Baldwin–Lomax (BL) turbulence models are analyzed with the present technique for the case of long slender blunt bodies. The Baldwin–Lomax turbulence model, which does not need the determination of the edge of the boundary layer, is modified in the present work to include the effects of variable properties and damping factor to accurately calculate flowfield characteristics in turbulent regions. Detailed comparisons are made with other numerical and experimental results to assess the accuracy and efficiency of the present solution procedure for computing hypersonic transitional/turbulent flows over long slender blunt bodies. The results of these computations are found to be in good agreement with available data. The effects of real gas on the flowfield characteristics are also studied.

Dual-code solution procedure for efficient computing equilibrium hypersonic axisymmetric laminar flows

An appropriate combination of the thin-layer Navier–Stokes (TLNS) and parabolized Navier–Stokes (PNS) solvers is used to accurately and efficiently compute hypersonic flowfields of equilibrium air around blunt-body configurations. The TLNS equations are solved in the nose region to provide the initial data plane needed for the solution of the PNS equations. Then the PNS equations are employed to efficiently compute the flowfield for the afterbody region by using a space marching procedure. Both the TLNS and PNS equations are numerically solved by using the efficient implicit non-iterative finite-difference algorithm of Beam and Warming. A shock fitting technique is used in both the TLNS and PNS codes to obtain accurate solution in the vicinity of the shock. In the regions of the flowfield with upstream influences, the globally iterated PNS (IPNS) equations, which are computationally more efficient than the TLNS equations, can be used to improve the accuracy of the results. To validate the results of the developed TLNS code and study real gas effects on the flowfield, hypersonic laminar flow over a sphere at Mach number of 11.26 is computed. The computations are performed for hypersonic laminar flow over 5° long slender blunt cone at Mach number of 19.25 to demonstrate the accuracy and efficiency of using the present TLNS-PNS methodology. To show the capability and robustness of the present solution procedure, the comparisons are also made with the wall heat-transfer data on a 12.84 ° / 7 ° bicone at Mach number of 6.89 and freestream temperature of 1604 K. The results of these computations are found to be in good agreement with available numerical and experimental data. The results of the perfect gas for these cases are also presented and the effects of real gas on the flowfield characteristics are studied in both the TLNS and PNS solutions.

Numerical simulation of stage separation manoeuvre with jet interaction

PurposeThis research aims to investigate numerically the influence of braking jets on separation process of a flying object.Design/methodology/approachThe flying object is at supersonic regime and axial separation of its stages is accomplished with the aid of braking jets of separated stage. The simulation is three‐dimensional and relative motion of the stages is considered three degree of freedom. Full Navier‐Stokes equations in conjunction with κ−ε (RNG) turbulence model equations are considered as governing equations. These equations are solved using the finite volume technique. The separation process is analysed as an unsteady process and the problem is solved in a moving grid domain. The local remeshing method is adapted to regenerate computational cells around moving boundaries.FindingsTime history of flow field around the vehicle components, time history of aerodynamic coefficients, and instantaneous relative position of the body components are the results of this research. Numerical modelling results are compared with the results of other references.Originality/valueMost of the similar works in this area have used Euler or thin layer Navier‐Stokes (TLNS) equations as governing equations and the use of full Navier‐Stokes equations to analyse such a complicated problem (3D axial separation with braking jets) have not been reported in the literature. Since there are some recirculation zones inside the flow field and Euler or even TLNS equations cannot predict their behaviours, the use of full Navier‐Stokes equations may lead to more accurate prediction of these regions and aerodynamic forces.

Aileron Buzz Simulation Using an Implicit Multiblock Aeroelastic Solver

A fully implicit multiblock aeroelastic solver, which coupled thin-layer Navier-Stokes equations with structural equations of motion, has been developed for the flutter simulation on complex aerodynamic configurations. Navier-Stokes equations are solved with a lower-upper symmetric Gauss-Seidel subiteration algorithm and a modified Harten-Lax-van Leer-Einfeldt-Wada scheme. Structural equations of motion are discretized by a direct second-order differential method with subiteration in generalized coordinates. The transfinite interpolation is used for the grid deformation of the blocks neighboring the flexible surface. To evaluate the effectiveness of the grid deformation, a comparison is first done for the oscillating LANN wing with a rigidly attached grid and a deforming grid. Then the method is applied to predict the flutter speed and frequency of the AGARD 445.6 standard aeroelastic wing. Finally, the aeroelastic instability referred to as aileron buzz is simulated for the supersonic transport model of the National Aerospace Laboratory of Japan.

Numerical Analysis of Dynamic Stability of a Reentry Capsule at Transonic Speeds

Dynamic stability of a reentry capsule in transonic speeds is discussed. An unsteady flowfield around the capsule under the forced pitching oscillation in the transonic flow of M = 1.3 is numerically simulated based on the three-dimensional thin-layer Navier-Stokes equations. The numerical result reveals that the dynamic instability is caused by the phase delay of the base pressure. It is also found that the base pressure, the recompression shock wave, and the wake behind the recompression shock wave all oscillate with the same delay time. The flow mechanism is proposed based on the idea that the phase delay of the base pressure is caused by a feedback loop of the flowfield behind the capsule. This flow mechanism reasonably explains the features observed in the present numerical simulation, as well as the experimental fact that the dynamic instability occurs at very low reduced frequencies.

Multidisciplinary optimization for gas turbine airfoil design

A multidisciplinary optimization procedure for gas turbine airfoil design has been developed and demonstrated on a generic blade. Aerodynamic and heat transfer design objectives are integrated along with various constraints on the airfoil geometry. Heat transfer optimization and external aerodynamic shape optimization has been performed on a generic blade. Shape optimization is performed using geometric parameters associated with film cooling and blade external shape. The blade is cooled both internally and externally (film cooling). Thin layer Navier-Stokes equations are used for the external flow field evaluations and finite element analysis is used to determine the blade interior temperatures. Total pressure loss, exit kinetic energy loss, average blade temperature and maximum blade temperature are minimized, with constraints on the blade geometry. The constrained multiobjective optimization problem is solved using the Kreisselmeier-Steinhauser (K-S) function approach. The results for the generic blade demonstrate the effectiveness of the optimization procedure.

Numerical Solution of Incompressible Unsteady Flows in Turbomachinery

A three-dimensional incompressible viscous flow solver of the thin-layer Navier-Stokes equations was developed for the unsteady turbomachinery flow computations. The solution algorithm for the unsteady flows combines the dual time stepping technique with the artificial compressibility approach for solving the incompressible unsteady flow governing equations. For time accurate calculations, subiterations are introduced by marching the equations in the pseudo-time to fully recover the incompressible continuity equation at each real time step, accelerated with a multi-grid technique. Computations of test cases show satisfactory agreements with corresponding theoretical and experimental results, demonstrating the validity and applicability of the present method to unsteady incompressible turbomachinery flows.

Viscous simulation of F-16 like configuration at high AoA

A viscous simulation of the flow over an F-16A model with inlet and the ventral fins, which were normally not considered in previous studies, is described. The method uses a patch grid multi-block, multi-grid Navier-Stokes solver. A novel method to grid the intake and diverter is used. The arrangement of the gridding system adopted minimises the number of grids needed particularly for viscous computations. A modified form of the Baldwin-Lomax algebraic turbulence model was used for the high AoA studies. To simplify the incorporation of turbulence model, the blocking arrangement is such that it allows a thin-layer Navier-Stokes equation formulation in the calculation of the eddy viscosity. A series of four flow conditions were studied; M = 0.9, AoA = 4°, 6.0°; and M = 1.2 at AoA = 6.0°; and M = 1.85 at AoA = 16°. Extensive comparisons with experimental data of the pressure distribution on the wing and on different streamwise stations show the modelling to be reasonably accurate. For the results presented here, the wing tip missile is ignored.

Effect of heat transfer on periodic transonic flows

AbstractThe effects of the model surface to free stream adiabatic temperature ratio (Tw/Tad) on periodic transonic flow over a 14% thick biconvex aerofoil are evaluated using a computational fluid dynamic approach. The analysis is based on the thin layer Navier Stokes equations with Baldwin-Lomax turbulence model. The results of computations showed that on biconves aerofoils there is a large effect of heat transfer on instantaneous pressure distributions and periodic buffet excitation level confirming some of the available experimental data. The effects observed have an implication in wind tunnel measurement of buffet associated periodic transonic flows.

Application of the reduced Navier–Stokes methodology to flow stability of Falkner–Skan class flows

This investigation ascertains the ability of the reduced Navier–Stokes (RNS) methodology to model linear flow stability. This is accomplished through development and investigation of two reduced forms of the Orr–Sommerfeld equation and of a second order RNS direct numerical simulation (DNS). The stability of five Falkner–Skan flows ( β=1.0, 0.2, 0.0, −0.1, and −0.1988) is investigated for these modified forms of the Orr–Sommerfeld equation (OSE). Neutral stability curves are numerically generated and compared for three forms of the OSE, viz. full Navier–Stokes equations, two-dimensional thin-layer Navier–Stokes equations which exclude only axial diffusion, and two-dimensional reduced Navier–Stokes equations which exclude all axial diffusion, as well as all diffusion in the normal momentum equation. Effects of a deferred corrector to include these terms are also investigated. Results of the computations demonstrate that the reduced forms of the OSE are consistent with the full OSE. With confirmation that the reduced Navier–Stokes equations contain the information required to properly model flow stability, development of a new class of asymptotic theories, stability methods, and approaches to direct numerical simulations, based on the RNS methodology, becomes feasible. Results from full DNS calculations using the RNS equations demonstrate the proper characteristics for disturbance growth and decay of the velocity disturbances. Velocity disturbance profiles are also of the required shape and magnitude.

Unsteady flow around helicopter rotor blade sections in forward flight

AbstractThe aerodynamic performance of aerofoils performing unsteady motions is important for the design of helicopter rotors. In this respect the study of aerofoils undergoing in-plane oscillations (translation along the horizontal axis) provides useful insight into the flow physics associated with the advancing blade in forward flight. In this paper a numerical method is developed in which the unsteady thin layer Navier-Stokes equations are solved for aerofoils performing rigid body motions. The method has been applied to the calculation of the flowfield around a NACA 0012 aerofoil performing in-plane motions representative of high-speed forward flight. Comparison of computed pressure data with experimental measurements is generally found to be good. The quantitative differences observed between computations and experiment are thought to have arisen mainly as a consequence of the low aspect ratio of the model rotor employed in the windtunnel tests.

Computational Investigation of Wing Flutter

Numerical aeroelastic simulation of a high-aspect-ratio transport type wing model in transonic region is presented. The aeroelastic responses of the wing are extracted by integrating compressible thin-layer Navier-Stokes equations coupled with the equations of motion of the wing structure, in a time dependent manner. The Yee-Harten implicit TVD scheme and the Wilson's θ method are employed to integrate these equations, respectively. Flutter boundaries were found for Mach number range, 0.7 to 0.85 and the results were compared with experimental flutter boundaries. Futhermore, Limit Cycle Oscillations were found and the characteristics of the flutter and limit cycle oscillations are investigated and discussed.

A multigrid Navier-Stokes CFD code for rotor computations

This paper presents the development of a multigrid Navier-Stokes code TLNS3DR for rotary wing calculations. The TLNS3DR is developed from the fixed-wing TLNS3D code by including a rotation term and making necessary modifications on boundary conditions. The TLNS3D code solves the thin-layer Navier-Stokes equations using an explicit multistage Runge-Kutta type of time stepping with multigrid method. The TLNS3DR code reformulates the TLNS3D code in the rotor blade-fixed moving frame of reference and solves relative motion of fluids. In this paper, the formulation of the Navier-Stokes equations in the moving frame of reference in terms of relative velocity is given. Several numerical examples are then presented. The results of rotor blade in hover calculated by the new TLNS3DR code seem quantitative correct. The computation also shows that this multigrid code is efficient.

A Computational Fluid Dynamics and Heat Transfer Model for Gaseous Core and Very High Temperature Gas-Cooled Reactors

A computational approach to the solution of Navier-Stokes equations for the thermal and flow fields of very high temperature gas-cooled and gaseous core reactors is presented. An implicit-explicit, finite volume, MacCormack method, in conjunction with the Gauss-Seidel line iteration procedure, is utilized to solve axisymmetric, thin-layer Navier-Stokes equations. An enthalpy rebalancing scheme is implemented to allow the convergence solutions to be obtained with the application of a wall heat flux. The subsonic and supersonic flows of helium in a very high temperature gas-cooled reactor and uranium tetrafluoride (UF4) in a gaseous core reactor under variable boundary conditions (such as adiabatic, isothermal, and constant heat flux) are calculated. The numerical results are compared with other published results and experimental-based correlations. The good agreement with empirical correlations indicates the usefulness of the presented model for the prediction of the flow and temperature distribution under the convective and radiative heat transfer environment of very high temperature gas-cooled and gaseous core reactors.

TWO-DIMENSIONAL TRANSONIC AEROELASTIC ANALYSIS USING THIN-LAYER NAVIER–STOKES METHOD

A robust efficient upwind implicit time-marching algorithm for the two-dimensional unsteady Thin-Layer Navier–Stokes equations, employing subiterations, is presented, especially directed to the aeroelastic analysis in viscous transonic flow. The purpose of using subiterations is to accelerate steady-state convergence and to permit a large time step in time-accurate simulations, thereby reducing the computational cost, while maintaining adequate accuracy. The ability of the method is demonstrated for the cases of inherently unsteady flow due to shock-induced separation, forced vibrations with large time steps,O(10) per period, and an aeroelastic analysis of a two-degree-of-freedom airfoil.

Wing Section Optimization for Supersonic Viscous Flow

To improve the performance of a highly swept supersonic wing, it is desirable to have an automated design method that also includes a higher fidelity to the flow physics. With this impetus, an aerodynamic optimization methodology incorporating the thin-layer Navier-Stokes equations and sensitivity analysis had previously been developed. Prior to embarking upon the full wing design task, the present investigation concentrated on the identification of effective optimization problem formulations and testing the feasibility of the employed methodology, by defining two-dimensional test cases. Starting with two distinctly different initial airfoils, two independent optimizations resulted in shapes with similar features: cambered, parabolic profiles with sharp leading- and trailing-edges. Secondly, an outboard wing section normal to the subsonic portion of the leading edge, which had a high normal angle-of attack, was considered. The optimization resulted in a shape with twist and camber that eliminated the adverse pressure gradient, hence, exploiting the leading-edge thrust. The wing section shapes obtained in all the test cases included the features predicted by previous studies. This was considered as a strong indication that the flow field analyses and sensitivity coefficients were computed and provided to the present gradient-based optimizer correctly. Also, from the results of the present study, effective optimization problem formulations could be deduced to start a full wing shape optimization.

Two-Dimensional Computational Model for Wave Rotor Flow Dynamics

A two-dimensional (θ, z) Navier–Stokes solver for multiport wave rotor flow simulation is described. The finite-volume forms of the unsteady thin-layer Navier–Stokes equations are integrated in time on multiblock grids that represent the stationary inlet and outlet ports and the moving rotor passages of the wave rotor. Computed results are compared with three-port wave rotor experimental data. The model is applied to predict the performance of a planned four-port wave rotor experiment. Two-dimensional flow features that reduce machine performance and influence rotor blade and duct wall thermal loads are identified.

ナビエ・ストークス方程式に基づく高アスペクト比後退翼のフラッタシミュレーション

Numerical flutter simulation of a high-aspect-ratio transport type of wing in transonic region is presented. The aeroelastic responses of the wing are obtained by integrating compressible thin-layer Navier-Stokes (N-S) equations coupled with the equations of motion of the wing structure. The Yee-Harten implicit TVD scheme and the Wilson θ method are employed to integrate N-S equations and structural ones, respectively. The flutter boundaries are obtained at the angle of attack of 2 degrees and compared with experimental results. The flutter characteristics of this wing model are investigated and discussed. Furthermore, Limit cycle oscillations are found in the unstable region and the mechanism is clarified.

Development of square nozzles for supersonic low-disturbance wind tunnels

Two Mach 2.4 nozzles with square test sections have been designed and analyzed, as part of an effort to develop low-disturbance facilities for laminar-flow control for high-speed civil transport. The mean flows have been simulated using a finite volume, central-differencing scheme to solve the thin-layer NavierStokes equations. Substantial crossflow exists in the boundary layers of both nozzles. The crossflow changes direction about halfway between the throat and exit, because of a change in the sign of the crossflow pressure gradient. S-shaped crossflow profiles are present in this region, just as on a swept wing. Both the standard crossflow Reynolds number and the new Reed and Haynes transition estimator predict transition to occur in the throat as well as near the exit. An analysis of the crossflow pressure gradients in the transonic throat region indicates that streamwise curvature has only a weak effect on the crossflow pressure gradient. Further research on crossflow-induced transition will have to be carried out before these nozzles will be suitable for low Mach number quiet-tunnel designs.