Showing papers on "Navier–Stokes equations published in 1991"

A dynamic subgrid‐scale eddy viscosity model

Abstract: One major drawback of the eddy viscosity subgrid‐scale stress models used in large‐eddy simulations is their inability to represent correctly with a single universal constant different turbulent fields in rotating or sheared flows, near solid walls, or in transitional regimes. In the present work a new eddy viscosity model is presented which alleviates many of these drawbacks. The model coefficient is computed dynamically as the calculation progresses rather than input a priori. The model is based on an algebraic identity between the subgrid‐scale stresses at two different filtered levels and the resolved turbulent stresses. The subgrid‐scale stresses obtained using the proposed model vanish in laminar flow and at a solid boundary, and have the correct asymptotic behavior in the near‐wall region of a turbulent boundary layer. The results of large‐eddy simulations of transitional and turbulent channel flow that use the proposed model are in good agreement with the direct simulation data.

Low‐dimensional models for complex geometry flows: Application to grooved channels and circular cylinders

Abstract: Two‐dimensional unsteady flows in complex geometries that are characterized by simple (low‐dimensional) dynamical behavior are considered. Detailed spectral element simulations are performed, and the proper orthogonal decomposition or POD (also called method of empirical eigenfunctions) is applied to the resulting data for two examples: the flow in a periodically grooved channel and the wake of an isolated circular cylinder. Low‐dimensional dynamical models for these systems are obtained using the empirically derived global eigenfunctions in the spectrally discretized Navier–Stokes equations. The short‐ and long‐term accuracy of the models is studied through simulation, continuation, and bifurcation analysis. Their ability to mimic the full simulations for Reynolds numbers (Re) beyond the values used for eigenfunction extraction is evaluated. In the case of the grooved channel, where the primary horizontal wave number of the flow is imposed from the channel periodicity and so remains unchanged with Re, the models extrapolate reasonably well over a range of Re values. In the case of the cylinder wake, however, due to the significant spatial wave number changes of the flow with the Re, the models are only valid in a small neighborhood of the decompositional Reynolds number.

Steady and unsteady solutions of the incompressible Navier-Stokes equations

Abstract: An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The equations are solved with a line-relaxation scheme that allows the use of very large pseudotime steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. The steady-state solution of flow through a square duct with a 90-deg bend is computed, and the results are compared with experimental data. Good agreement is observed. Computations of unsteady flow over a circular cylinder are presented and compared to other experimental and computational results. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.

Exact Solutions of the Steady-State Navier-Stokes Equations

Abstract: 1. The solutions represent fundamental fluid-dynamic flows. Also, owing to the uniform validity of exact solutions, the basic phenomena described by the Navier-Stokes equations can be more closely studied. 2. The exact solutions serve as standards for checking the accuracies of the many approximate methods, whether they are numerical, asymp­ totic, or empirical. Current advances in computer technology make the complete numerical integration of the Navier-Stokes equations more feasible. However, the accuracy of the results can only be ascertained by a comparison with an exact solution.

Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes

Abstract: This paper treats the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. (For example, in the three-dimensional case, the obstacles have a size of e3 and are located at the nodes of a regular mesh of size e.) A suitable extension of the pressure is used to prove the convergence of the homogenization process to a Brinkman-type law (in which a linear zero-order term for the velocity is added to a Stokes or Navier-Stokes equation).

Tip Leakage Flow in Axial Compressors

Abstract: Experimental measurements in a linear cascade with tip clearance are complemented by numerical solutions of the three-dimensional Navier–Stokes equations in an investigation of tip leakage flow. Measurements reveal that the clearance flow, which separates near the entry of the tip gap, remains unattached for the majority of the blade chord when the tip clearance is similar to that typical of a machine. The numerical predictions of leakage flow rate agree very well with measurements, and detailed comparisons show that the mechanism of tip leakage is primarily inviscid. It is demonstrated by simple calculation that it is the static pressure field near the end of the blade that controls chordwise distribution of the flow across the tip. Although the presence of a vortex caused by the roll-up of the leakage flow may affect the local pressure field, the overall magnitude of the tip leakage flow remains strongly related to the aerodynamic loading of the blades.

Vortex Element Methods for Fluid Dynamic Analysis of Engineering Systems: Vortex cloud modelling by the boundary integral method

Abstract: Part I. The Surface Vorticity Boundary Integral Method of Fluid Flow Analysis: 1. The basis of surface singularity modelling 2. Lifting bodies, two-dimensional aerofoils and cascades 3. Mixed-flow and radial cascades 4. Bodies of revolution, ducts and annuli 5. Ducted propellers and fans 6. Three-dimensional and meridional flows in turbo-machines 7. Free vorticity shear layers and inverse methods Part II. Vortex Dynamics and Vortex Cloud Analysis: 8. Vortex dynamics in inviscid flows 9. Simulation of viscous diffusion in discrete vortex modelling 10. Vortex cloud modelling by the boundary integral method 11. Further development and applications of vortex cloud modelling to lifting bodies and cascades 12. Use of grid systems in vortex dynamics and meridional flows Appendix.

The failure on continuous dependence on initial data for the Navier-Stokes equations of compressible flow

Abstract: It is shown that physical solutions of the Navier–Stokes equations for one-dimensional, compressible flow need not depend continuously on their initial data, at least when vacuum states are allowed. Specifically, two fluid regions initially separated by a third region of very low density 6 are considered. It is shown that, as $\delta \to 0$, the (unique) solutions corresponding to$\delta > 0$ do not in fact converge to a physical solution, but rather to a nonphysical weak solution in which the two fluids cannot collide, independent of their initial velocities, and whose separate momenta need not be conserved. A particular consequence is that solutions of the cavity problem $\delta = 0$ are not unique.

Homogenization of the Navier-Stokes Equations in Open Sets Perforated with Tiny Holes II: Non-Critical Sizes of the Holes for a Volume Distribution and a Surface Distribution of Holes

Abstract: This paper is devoted to the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. For obstacles of critical size it was established in Part I that the limit problem is described by a law of Brinkman type. Here we prove that for smaller obstacles, the limit problem reduces to the Stokes or Navier-Stokes equations, and for larger obstacles, to Darcy's law. We also apply the abstract framework of Part I to the case of a domain containing tiny obstacles, periodically distributed on a surface. (For example, in three dimensions, consider obstacles of size e2, located at the nodes of a regular plane mesh of period e.) This provides a mathematical model for fluid flows through mixing grids, based on a special form of the Brinkman law in which the additional term is concentrated on the plane of the grid.

Numerical Simulation of Hydraulic Jump

Abstract: Boussinesq equations describing one-dimensional unsteady, rapidly varied flows are integrated numerically to simulate both the sub- and supercritical flows and the formation of a hydraulic jump in a rectangular channel having a small bottom slope. For this purpose the MacCormack (second-order accurate in space and time) and two-four (second-order accurate in-time and fourth-order in space) explicit finite-difference schemes are used to solve the governing equations subject to specified end conditions until a steady state is reached. The inclusion of initial and boundary conditions is discussed, and the importance of the Boussinesq terms is investigated. Complete test results for a range of Froude numbers are presented that may be used by other researchers for the verification of mathematical models. A comparison of the computed measured results shows that the agreement between them is satisfactory for the fourth-order finite-difference scheme although the second-order scheme does not accurately predict the location of the jump. These simulations show that the Boussinesq terms have little effect in determining the location of the hydraulic jump.

Aeroelastic Analysis of Wings Using the Euler Equations with a Deforming Mesh

Abstract: Modifications to the CFL3D three-dimensional unsteady Euler/Navier-Stokes code for the aeroelastic analysis of wings are described. The modifications involve including a deforming mesh capability that can move the mesh to continuously conform to the instantaneous shape of the aeroelastically deforming wing and also including the structural equations of motion for their simultaneous time integration with the governing flow equations. Calculations were performed using the Euler equations to verify the modifications to the code and as a first step toward aeroelastic analysis using the Navier-Stokes equations. Results are presented for the NACA 0012 airfoil and a 45-deg sweptback wing to demonstrate applications of CFL3D for generalized force computations and aeroelastic analysis. Comparisons are made with published Euler results for the NACA 0012 airfoil and with experimental flutter data for the 45-deg sweptback wing to access the accuracy of the present capability. These comparisons show good agreement and, thus, the CFL3D code may be used with confidence for aeroelastic analysis of wings. The paper describes the modifications that were made to the code and presents results and comparisons that assess the capability.

The response of isotropic turbulence to isotropic and anisotropic forcing at the large scales

Abstract: The nonlinear interscale couplings in a turbulent flow are studied through direct numerical simulations of the response of isotropic turbulence to isotropic and anisotropic forcing applied at the large scales. Specifically, forcing is applied to the energy‐containing wave‐number range for about two eddy turnover times to fully developed isotropic turbulence at Taylor‐scale Reynolds number 32 on an 1283 grid. When forced isotropically, the initially isotropic turbulence remains isotropic at all wave numbers. However, anisotropic forcing applied through an array of counter‐rotating rectilinear vortices induces high levels of anisotropy at the small scales. At low wave numbers the force term feeds energy directly into two velocity components in the plane of the forced vortices. In contrast, at high wave numbers the third (spanwise) component receives the most energy, producing small‐scale anisotropy very different from that at the large scales. Detailed analysis shows that the development of small‐scale anisotropy is caused primarily by nonlocal wave‐vector triads with one leg in the forced low‐wave‐number range. This latter result is particularly significant because asymptotic analysis of the Fourier‐transformed Navier–Stokes equations shows that distant triadic interactions coupling the energy‐containing and dissipative scales persist at asymptotically high Reynolds numbers, suggesting that the structural couplings between large and small scales in these moderate Reynolds number simulations would also exist in high Reynolds number forced turbulence. The results therefore imply a departure from the classical hypothesis of statistical independence between large‐ and small‐scale structure and local isotropy.

Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls

Abstract: We examine certain analytic and numerical aspects of optimal control problems for the stationary Navier-Stokes equations. The controls considered may be of either the distributed or Neumann type; the functionals minimized are either the viscous dissipation or the L4_distance of candidate flows to some desired flow. We show the existence of optimal solutions and justify the use of Lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced. We study the regularity of solutions of this system. Then, we consider the approximation, by finite element methods, of solutions of the optimality system and derive optimal error estimates.

Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space

Abstract: The existence of time-periodic solutions to the system governing the motion of an incompressible fluid filling the whole space is proved. It is assumed that the force field acting over the fluid is periodic on time. Moreover, uniqueness and stability of this solution with respect to a suitable norm is obtained.

A Numerical Study of Developing Free Convection Between Isothermal Vertical Plates

Abstract: Steady two-dimensional laminar free convection between isothermal vertical plates including entrance flow effects has been numerically investigated. The full elliptic forms of the Navier-Stokes and energy equations are solved using novel inlet flow boundary conditions. Results are presented for Prandtl number Pr = 0.7, Grashof number range 50 {le} Gr{sub b} {le} 5 {times} 10{sup 4}, and channel aspect ratios of L/b = 10, 17, 24. New phenomena, such as inlet flow separation, have been observed. The results cast doubt on the validity of previous elliptic solutions. Comparisons with the approximate boundary-layer results show that a full elliptic solution is necessary to get accurate local quantities near the channel entrance.

Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary

Abstract: We prove the convergence of the homogenization process for a nonstationary Navier-Stokes system in a porous medium. The result of homogenization is Darcy's law, as in the case of the Stokes equation, but the convergence of pressures is in a different function space.