Showing papers on "Incompressible flow published in 1994"

Vorticity and turbulence

Abstract: This is an introduction to turbulence in vortex systems, and to turbulence theory for incompressible flow described in terms of the vorticity field. It is the author's hope that by the end of the book the reader will believe that these subjects are identical, and constitute a special case of fairly standard statistical mechanics, with both equilibrium and non-equilibrium aspects. The author's main goal is to relate turbulence to statistical mechanics. The first three chapters of the book constitute a fairly standard introduction to homogeneous turbulence in incompressible flow; a quick review of fluid mechanics; a summary of the appropriate Fourier theory; and a summary of Kolmogorov's theory of the inertial range. The next four chapters present the statistical theory of vortex notion, and the vortex dynamics of turbulence. The book ends with the major conclusion that turbulence can no longer be viewed as incomprehensible.

Effects of the computational time step on numerical solutions for turbulent flow

Abstract: Effects of large computational time steps on the computed turbulence were investigated using a fully implicit method. In turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined. Turbulence fluctuations could not be sustained if the computational time step was near or larger than the Kolmogorov time scale.

Geometric statistics in turbulence

Abstract: The author presents results regarding certain average properties of incompressible fluids derived from the equations of motion. The author estimates the average dissipation rate, the average dimension of level sets. The role played by the field of direction of vorticity in the three-dimensional Euler and Navier-Stokes equations is discussed and a class of two-dimensional equations that are useful models of incompressible dynamics is described. The author presents results concerning scaling exponents in turbulence.

Optimal energy density growth in Hagen-Poiseuille flow

Abstract: Linear stability of incompressible flow in a circular pipe is considered. Use is made of a vector function formulation involving the radial velocity and radial vorticity only. Asymptotic as well as transient stability are investigated using eigenvalues and e-pseudoeigenvalues, respectively. Energy stability is probed by establishing a link to the numerical range of the linear stability operator. Substantial transient growth followed by exponential decay has been found and parameter studies revealed that the maximum amplification of initial energy density is experienced by disturbances with no streamwise dependence and azimuthal wavenumber n = 1. It has also been found that the maximum in energy scales with the Reynolds number squared, as for other shear flows. The flow field of the optimal disturbance, exploiting the transient growth mechanism maximally, has been determined and followed in time. Optimal disturbances are in general characterized by a strong shear layer in the centre of the pipe and their overall structure has been found not to change significantly as time evolves. The presented linear transient growth mechanism which has its origin in the non-normality of the linearized Navier–Stokes operator, may provide a viable process for triggering finite-amplitude effects.

An acoustic/viscous splitting technique for computational aeroacoustics

Abstract: A new technique for the numerical analysis of aerodynamic noise generation is developed. The approach involves first solving for the time-dependent incompressible flow for the given geometry. A “hydrodynamic” density correction to the constant incompressible density is then calculated from knowledge of the incompressible pressure fluctuations. The compressible flow solution is finally obtained by considering perturbations about the “corrected” incompressible flow. This fully nonlinear technique, which is tailored to extract the relevant acoustic fluctuations, appears to be an efficient approach to the numerical analysis of aerodynamic noise generation, particularly in viscous flows. Applications of this technique to some classical acoustic problems of interest, including some with moderately high subsonic Mach numbers, are presented to validate the approach. The technique is then applied to a fully viscous problem where sound is generated by the flow dynamics.

Eigenanalysis of unsteady flow about airfoils, cascades, and wings

Abstract: A general technique for constructing reduced order models of unsteady aerodynamic flows about twodimensional isolated airfoils, cascades of airfoils, and three-dimensional wings is developed. The starting point is a time domain computational model of the unsteady small disturbance flow. For illustration purposes, we apply the technique to an unsteady incompressible vortex lattice model. The eigenmodes of the system, which may be thought of as aerodynamic states, are computed and subsequently used to construct computationally efficient, reduced order models of the unsteady flowfield. Only a handful of the most dominant eigenmodes are retained in the reduced order model. The effect of the remaining eigenmodes is included approximately using a static correction technique. An important advantage of the present method is that once the eigenmode information has been computed, reduced order models can be constructed for any number of arbitrary modes of airfoil motion very inexpensively. Numerical examples are presented that demonstrate the accuracy and computational efficiency of the present method. Finally, we show how the reduced order model may be incorporated into an aeroelastic flutter model.

Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble fronts

Abstract: A potential flow model of Rayleigh–Taylor and Richtmyer–Meshkov bubbles on an interface between an incompressible fluid and a constant supporting pressure (Atwood number A=1) is presented. In the model, which extends the work of Layzer [Astrophys. J. 122, 1 (1955)], ordinary differential equations for the bubble heights and curvatures are obtained by considering the potential flow equations near the bubble tips. The model is applied to two‐dimensional single‐mode evolution as well as two‐bubble competition, for both the Rayleigh–Taylor (RT) and the Richtmyer–Meshkov (RM) instabilities, the latter treated in an impulse approximation. The model predicts that the asymptotic velocity of a single‐mode RM bubble of wavelength λ decays as λt−1, in contrast with the constant asymptotic velocity attained in the RT case. Bubble competition, which is believed to determine the multimode front evolution, is demonstrated for both the RT and RM instabilities. The capability of the model to predict bubble growth in a fin...

A study of the fine‐scale motions of incompressible time‐developing mixing layers

Abstract: The geometry of dissipating motions in direct numerical simulations (DNS) of the incompressible mixing layer is examined. All nine partial derivatives of the velocity field are determined at every grid point in the flow, and various invariants and related quantities are computed from the velocity gradient tensor. Motions characterized by high rates of kinetic energy dissipation and high enstrophy density are of particular interest. Scatter plots of the invariants are mapped out and interesting and unexpected patterns are seen. Depending on initial conditions, each type of shear layer produces its own characteristic scatter plot. In order to provide more detailed information on the distribution of invariants at intermediate and large scales, scatter plots are replaced with more useful number density contour plots. These essentially represent the unnormalized joint probability density function of the two invariants being cross‐plotted. Plane mixing layers at the same Reynolds number, but with laminar and turbulent initial conditions, are studied, and comparisons of the rate‐of‐strain topology of the dissipating motions are made. The results show conclusively that, regardless of initial conditions, the bulk of the total kinetic energy dissipation is contributed by intermediate scale motions, whose local rate‐of‐strain topology is characterized as unstable‐node‐saddle–saddle (two positive rate‐of‐strain eigenvalues, one negative). In addition, it is found that, for these motions, the rate‐of‐strain invariants tend to approximately follow a straight line relationship, characteristic of a two‐dimensional flow with out of plane straining. In contrast, fine‐scale motions, which have the highest dissipation, but which only contribute a small fraction of the total dissipation tend toward a fixed ratio of the principal rates of strain.

Variational bounds on energy dissipation in incompressible flows: Shear flow.

Abstract: A variational principle for upper bounds on the time averaged rate of viscous energy dissipation for Newtonian fluid flows is derived from the incompressible Navier-Stokes equations. When supplied with appropriate test background'' flow fields, the variational formulation produces explicit estimates for the energy dissipation rate. This dissipation rate is related to the drag of the fluid on the boundaries, and so these estimates translate into bounds on the drag. We analyze the problem of boundary-driven shear flow in detail, comparing the rigorous estimates obtained from the variational method with both recent experimental results and predictions of a conventional closure approximation from statistical turbulence theory.

On the generation of turbulent wall friction

Abstract: The formation of streaky velocity structures in the near wall region of turbulent boundary layers is studied through a simplified two‐dimensional computational model in the plane normal to the average velocity. It is shown that the redistribution of the longitudinal velocity by streamwise vortices produces features very similar to those observed in the experiments, and that compact streamwise vortices form naturally from more general vorticity distributions. It is also shown, both numerically and analytically, that one effect of the formation of the streaks is to increase the average wall friction, and it is suggested that this effect is responsible for the higher friction in turbulent boundary layers, as opposed to laminar ones. An approximate quantitative analysis of the process supports this assumption.

On passive scalar derivative statistics in grid turbulence

Abstract: The probability density function, and related statistics, of scalar (temperature) derivative fluctuations in decaying grid turbulence with an imposed cross‐stream, passive linear temperature profile, is studied for a turbulence Reynolds number range, Rel, varying from 50 to 1200, (corresponding to a Taylor Reynolds number range 30

On the role of linear mechanisms in transition to turbulence

Abstract: Recent work has shown that linear mechanisms can lead to substantial transient growth in the energy of small disturbances in incompressible flows even when the Reynolds number is below the critical value predicted by linear stability (eigenvalue) analysis. In this note it is shown that linear growth mechanisms are necessary for transition in flows governed by the incompressible Navier–Stokes equations and that non‐normality of the linearized Navier–Stokes operator is a necessary condition for subcritical transition.

Global solutions of two-dimensional Navier-Stokes and euler equations

Abstract: Long-time solutions to the Navier-Stokes (NS) and Euler (E) equations of incompressible flow in the whole plane are constructed, under the assumption that the initial vorticity is in L1(ℝ2) for (NS) and in L1(ℝ2)∩ Lr(ℝ2) for some r>2 for (E). It is shown that the solution to (NS) is unique, smooth and depends continuously on the initial data, and that the (velocity) solution to (E) is Holder continuous in the space and time coordinates. It is shown that as the viscosity vanishes, there is a subsequence of solutions to (NS) converging to a solution of (E).

A low‐dimensional Galerkin method for the three‐dimensional flow around a circular cylinder

Abstract: A low‐dimensional Galerkin method for the three‐dimensional flow around a circular cylinder is constructed. The investigation of the wake solutions for a variety of basic modes, Hilbert spaces, and expansion modes reveals general mathematical and physical aspects which may strongly effect the success of low‐dimensional simulations. Besides the cylinder wake, detailed information about the construction of similar low‐dimensional Galerkin methods for the sphere wake, the boundary‐layer, the flow in a channel or pipe, the Taylor–Couette problem, and a variety of other flows is given.

A numerical study of pressure fluctuations in three‐dimensional, incompressible, homogeneous, isotropic turbulence

Abstract: Pressure fluctuations in incompressible turbulence are studied by direct numerical simulations of the three‐dimensional (3‐D) Navier–Stokes equations. The pressure probability distribution function (PDF) is shown to have an exponential tail on the negative side, and to be independent of the Reynolds number for Reλ≲60. At higher Reynolds numbers, the low pressure part of the pressure PDF becomes super exponential. The joint PDFs of strain, vorticity, and pressure (considered pairwise) show a strong dissymmetry between positive and negative pressure fluctuations. The results obtained from the numerical solutions of the Navier–Stokes equations are compared with a Gaussian velocity field. The two statistical ensembles are shown to lead to quantitatively different results.

Computation of oscillating airfoil flows with one- and two-equation turbulence models

Abstract: The ability of one- and two-equation turbulence models to predict unsteady separated flows over airfoils is evaluated. An implicit, factorized, upwind-biased numerical scheme is used for the integration of the compressible, Reynolds-averaged Navier-Stokes equations. The turbulent eddy viscosity is obtained from the computed mean flowfield by integration of the turbulent field equations. One- and two-equation turbulence models are first tested for a separated airfoil flow at fixed angle of incidence. The same models are then applied to compute the unsteady flowfields about airfoils undergoing oscillatory motion at low subsonic Mach numbers. Experimental cases where the flow has been tripped at the leading-edge and where natural transition was allowed to occur naturally are considered. The more recently developed turbulence models capture the physics of unsteady separated flow significantly better than the standard kappa-epsilon and kappa-omega models. However, certain differences in the hysteresis effects are observed. For an untripped high-Reynolds-number flow, it was found necessary to take into account the leading-edge transitional flow region to capture the correct physical mechanism that leads to dynamic stall.

Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean

Abstract: The well-known quasigeostrophic system (QGS) for zero Rossby number flow has been used extensively in oceanography and meteorology for modeling and forecasting mid-latitude oceanic and atmospheric circulation. Formulation of QGS requires a (singular) perturbation expansion of a set of primitive equations at small Rossby number, and the quasigeostrophic equation expresses conservation of the zero-order potential vorticity of the flow. The formal expansion is justified by investigating the behavior of solutions of a set of primitive equations (PE) with a particular scaling, in the limit of zero Rossby number. This primitive model represents adiabatic, inviscid, incompressible flow with variable density and Coriolis force. Difficulties arise because PE, scaled for small Rossby number, contains unwanted solutions varying on a fast time scale with frequencies inversely proportional to the Rossby number. Without restrictions on the initial conditions, solutions of the scaled PE model do not necessarily converge...

Fast multigrid method for solving incompressible hydrodynamic problems with free surfaces

Abstract: We develop of a finite-volume multigrid Euler scheme for solving three-dimensional, fully nonlinear ship wave problems. The flowfield and the a priori unknown free surface location are calculated by coupling the free surface kinematic and dynamic equations with the equations of motion for the bulk flow. The evolution of the free surface boundary condition is linked to the evolution of the bulk flow via a novel iteration strategy that allows temporary leakage through the surface before the solution is converged. The method of artificial compressibility is used to enforce the incompressibility constraint for the bulk flow. A multigrid algorithm is used to accelerate convergence to a steady state

Transition to chaos in converging–diverging channel flows: Ruelle–Takens–Newhouse scenario

Abstract: Direct numerical simulations of the transition process from laminar to chaotic flow in converging–diverging channels are presented. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasiperiodic, and chaotic self‐sustained flow regimes. The numerical experiments reveal three distinct bifurcations as the Reynolds number is increased, each adding a new fundamental frequency to the velocity spectrum. In addition, frequency‐locked periodic solutions with independent but synchronized periodic functions are obtained. A scenario similar to the Ruelle–Takens–Newhouse scenario of the onset of chaos is verified in this forced convective open system flow. The results are illustrated for different Reynolds numbers using time‐velocity histories, Fourier power spectra, and phase space trajectories. The global structure of the self‐sustained oscillatory flow for a periodic regime is also discussed.

Cavity Flow Predictions Based on the Euler Equations

Abstract: An Euler solver based on artificial-compressibility and pseudo-time stepping is developed for the analysis of partial sheet cavitation in two-dimensional cascades and on isolated airfoils. The computational domain is adapted to the evolution of the cavity surface and the boundary conditions are implemented on the cavity interface. This approach enables the cavitation pressure condition to be incorporated directly without requiring the specification of the cavity length or the location of the inception point. Numerical solutions are presented for a number of two-dimensional cavity flow problems, including both leading edge cavitation and the more difficult mid-chord cavitation conditions. Validation is accomplished by comparing with experimental measurements and nonlinear panel solutions from potential flow theory. The demonstrated success of the Euler cavitation procedure implies that it can be incorporated in existing incompressible CFD codes to provide engineering predictions of cavitation. In addition, the flexibility of the Euler formulation may allow extension to more complex problems such as viscous flows, time-dependent flows and three-dimensional flows.

On the Influence of Time-Varying Flow Velocity on Unsteady Aerodynamics.

Abstract: The effects of a periodic free-stream velocity on the unsteady aerodynamics of an airfoil in incompressible flow are examined. Existing theories are reviewed, and their simplifications and limitations are properly identified. A new general aerodynamic theory for an airfoil undergoing a combination of harmonic pitching, plunging and fore-aft motion is presented. An extension to arbitrary free-stream velocity variations and arbitrary airfoil motion is also given. The theoretical results are validated against numerical predictions made by a modern Euler code.

Linear stability theory of two-layer fluid flow in an inclined channel

Abstract: The linear stability of the two‐layer flow of immiscible, incompressible fluids in an inclined channel is considered. In the long‐wave limit, mechanisms for linear instability, and the consequences of competition between mechanisms, are identified. For arbitrary wave numbers, air–water and olive oil–water systems are considered, in order to determine the influence of the channel thickness and the mean interfacial height on the stability of the flow. This paper characterizes those physical situations in which the primary instability is to long‐wave interfacial disturbances. The odd Orr–Sommerfeld shear mode within the water layer, which is necessarily stable in plane Poiseuille flow, is found to grow and even be the dominant mode of instability for the olive oil–water system. The consequences beyond linear stability are discussed.

Deterministic forcing of homogeneous, isotropic turbulence

Abstract: A deterministic low‐wave‐number forcing scheme designed to obtain statistically stationary homogeneous, isotropic turbulence in incompressible flows, and address criticisms of earlier schemes is proposed. Three‐dimensional turbulent kinetic energy spectra collapse well and are more consistent with the experimentally determined Kolmogorov coefficient. Spectra for unforced scalar fields at different Prandtl numbers advected by the forced velocity fields collapse under Batchelor scaling, and do not show as strong a low‐wave‐number anomaly as earlier simulations that use forcing on both the velocity and scalar fields.

Numerical Computations of Supersonic Base Flow with Special Emphasis on Turbulence Modeling

Abstract: : A zonal, implicit, time-marching Navier-Stokes computational technique has been used to compute the turbulent supersonic base flow over cylindrical afterbodies. A critical element of calculating such flows is the turbulence model. Various eddy viscosity turbulence models have been used in the base region flow computations. These models include two algebraic turbulence models and a two-equation k-epsilon model. The k-epsilon equations are developed in a general coordinate system and solved using an implicit algorithm. Calculations with the k-epsilon model are extended up to the wall. Flow field computations have been performed for a cylindrical afterbody at M = 2.46 and at angle of attack alpha = 0. The results are compared to the experimental data for the same conditions and the same configuration. Details of the mean flow field as well as the turbulence quantifies have been presented. In addition, the computed base pressure distribution has been compared with the experiment. In general, the k-epsilon turbulence model performs better in the near wake than the algebraic models and predicts the base pressure much better. Base flow, Base pressure, Turbulence models, Wake, Supersonic flow.

Nonlinear long-wave stability of superposed fluids in an inclined channel

Abstract: We consider the two-layer flow of immiscible, viscous, incompressible fluids in an inclined channel. We use long-wave theory to obtain a strongly nonlinear evolution equation which describes the motion of the interface. This equation includes the physical effects of viscosity stratification, density stratification, and shear. A weakly nonlinear analysis of this equation yields a Kuramoto–Sivashinsky equation, which possesses a quadratic nonlinearity. However, certain physical situations exist in two-layer flow for which modifications of the Kuramoto–Sivashinsky equation are physically pertinent. In particular, the presence of the second layer can mediate the wave-steepening instability found in single-phase falling films, requiring the inclusion of a cubic nonlinearity in the weakly nonlinear analysis. The introduction of the cubic nonlinearity destroys the symmetry-breaking bifurcations of the Kuramoto–Sivashinsky equation, and new isolated solution branches emerge as the strength of the cubic nonlinearity increases. Bistability between these new solutions and those associated with the Kuramoto–Sivashinsky equation is found, as well as the formation of a hysteresis loop from smaller-amplitude travelling waves to larger-amplitude travelling waves. The physical implications of these dynamics to the phenomenon of laminar flooding in a channel are discussed.