Showing papers on "K-epsilon turbulence model published in 1976"

Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence

Abstract: The problem of closing the Reynolds-stress and dissipation-rate equations at low Reynolds numbers is considered, specific forms being suggested for the direct effects of viscosity on the various transport processes. By noting that the correlation coefficient is nearly constant over a considerable portion of the low-Reynolds-number region adjacent to a wall the closure is simplified to one requiring the solution of approximated transport equations for only the turbulent shear stress, the turbulent kinetic energy and the energy dissipation rate. Numerical solutions are presented for turbulent channel flow and sink flows at low Reynolds number as well as a case of a severely accelerated boundary layer in which the turbulent shear stress becomes negligible compared with the viscous stresses. Agreement with experiment is generally encouraging.

Spatially decaying turbulence and its relation to mixing across density interfaces

Abstract: The turbulence generated by a vertically oscillating grid in a water tank and the entrainment across a salinity interface caused by this turbulence have been investigated experimentally. Measurements were carried out in a homogeneous layer of fluid as well as a two-layered fluid, which permitted us to determine the decay law of this turbulence and the way in which the structure of the turbulence depends on the mesh size and on the frequency and amplitude of the grid oscillation. It was found that the turbulent kinetic energy decays with distance from the grid according to a power law , the Peclet number being high. While the bearing of these results on the problem of the thermocline or an inversion is clear we wish to emphasize that the spatial decay of turbulence is interesting in itself.

The calculation of near-wake flows

Abstract: Calculated flow properties are compared with measurements obtained in twodimensional isothermal wakes with and without recirculation. The equations of continuity and momentum were solved numerically together with equations which formed a turbulence model. Calculations were made using three turbulence models : the first comprised transport equations for turbulence kinetic energy and the rate of turbulence dissipation; the second and third comprised equations for the rate of turbulence dissipation and two forms of Reynolds-stress equations characterized by different redistribution terms. The results show that, for wakes without recirculation, the particular turbulence model is less important than the boundary condition assumed in the plane of the trailing edge of the body; though the Reynolds-stress models do, of course, provide a better representation of the individual normal stresses. In the case of wakes with recirculation, both the length of the recirculation region and the rate of spread of the downstream wake are underestimated. The second discrepancy is particularly evident and appears to stem from the form of the dissipation equation. A suggestion for improving the modelling of this equation is provided together with necessary justification.

Modulation of Turbulence Energy by Longitudinal Rolls in an Unstable Planetary Boundary Layer

Abstract: Horizontal roll vortices influence the distribution of turbulence, with turbulence variances and fluxes concentrated in regions of positive roll vertical velocity ωr. This “modulation” of turbulence can be explained simply in terms of the advection of turbulence-generating elements by rolls. A budget equation is derived for the roll-modulated turbulence energy. Evaluations of various terms in the equation shows that the modulation of turbulence variance is accounted for primarily by a similar modulation in mechanical and buoyancy production near the surface and by vertical transport at higher levels (∼100 m). Energy exchange between rolls and turbulence is relatively unimportant. That is, the rolls modulate, turbulence energy mainly by redistributing turbulence and turbulence-producing elements, rather than by exchanging energy. Similarly, it is shown that the exchange of energy between rolls and roll-modulated turbulence contributes considerably less to the energy equation of rolls than does the...

Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field

Abstract: The transition of homogeneous turbulence from an initially isotropic three-dimensional to a quasi-two-dimensional state is simulated numerically for a conducting, incompressible fluid under a uniform magnetic field B 0 . The magnetic Reynolds number is assumed to be small, so that the induced fluctuations of the magnetic field are small compared with the imposed magnetic field B 0 , and can be computed from a quasi-static approximation. If the imposed magnetic field is strong enough, all variations of the flow field in the direction of B 0 are damped out. This effect is important e.g. in the design of liquid-metal cooling systems for fusion reactors, and the properties of the final state are relevant to atmospheric turbulence. An extended version of the code of Orszag and Patterson (1972) is used to integrate the Navier-Stokes equations for an incompressible fluid. The initial hydrodynamic Reynolds number is 60. The magnetic interaction number N is varied between zero and 50. Periodic boundary conditions are used. The resolution corresponds to 323 points in real space. The full nonlinear simulations are compared with otherwise identical linear simulations; the linear results agree with the nonlinear ones within 3% for about one-fifth of the large-scale turnover time. This departure is a consequence of the return-to-equilibrium tendencies caused mainly by energy transfer towards high wavenumbers. The angular energy transfer and the energy exchange between different components are smaller, and become virtually zero for large values of N. For N [approximate] 50 we reach a quasi-two-dimensional state. Here, the energy transfer towards high wavenumbers is reduced for the velocity components perpendicular to B 0 but relatively increased for the component parallel to B 0 . The overall behaviour is more similar to three-than to purely two-dimensional turbulence. This finding is of great importance for turbulence models of the atmosphere. The realization of a purely two-dimensional state does not seem to be possible for decaying turbulence. The magnetic field causes highly intensified pressure fluctuations, which contribute to the redistribution of the anisotropic Lorentz forcing.

Computation of Separated Transonic Turbulent Flows

Abstract: The two-dimensional Reynolds averaged compressible Navier-Stokes equations are solved using MacCormack's second-order accurate explicit finite difference method to simulate the separated transonic tur- bulent flowfield over an airfoil. Four different algebraic eddy viscoisity models are tested for viability to achieve turbulence closure for the class of flows considered. These models range from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region. Results of this study indicate the necessity for special attention to the separated flow region and suggest limits of applicability of algebraic turbulence models to these separated flowfield. each of these studies the time-dependent Reynolds averaged Navier-Stokes equations for two-dimensional compressive flow are used and tur- bulence closure is achieved by means of model equations for the Reynolds stresses. Wilcox1'2 used a first-order accurate numerical scheme and the two equation differential tur- bulence model of Saffman 12 to simulate the supersonic shock boundary-layer interaction experiment of Reda and Mur- phy 13 and the compression corner flow of Law.14 Good quan- titative agreement with the Reda and Murphy data was ob- tained, but only the qualitative features of the compression corner flow were well simulated. Using a more sophisticated second-order accurate numerical scheme, Baldwin3'4 con- sidered both the two equation differential model of Saffman and a simpler algebraic mixing-length model to simulate the hypersonic shock boundary-layer interaction experiment of Holden.15 He found the more elaborate model of Saffman to yield somewhat better results than the algebraic model, but at the cost of considerably more computing time. Good quan- titative agreement with experiment was not obtained with either model. Following Baldwin's approach all subsequent investigations have been performed using the more rigorous second-order accurate numerical scheme of Mac- Cormack.17'18 Deiwert5'6'11 considered an algebraic mixing- length model to simulate the transonic airfoil experiment of McDevitt et al. 16 while Horstman et al. 8 used a similar ap- proach to simulate their hypersonic shock boundary-layer ex- periment on an axisymmetric cylinder. In each of these studies, while qualitative features of the flows were described well, good quantitative agreement with experiment in the in- teraction regions was not obtained. Using a relaxing turbulence model Shang and Hankey7 simulated the compression corner flow of Law, and Baldwin and Rose10 simulated the flat plate flow of Reda and Murphy. In each of these studies the relaxing model was found to per- form significantly better than the simpler algebraic model and, according to Shang and Hankey, provided significantly better comparisons with measurements than were obtained by Wilcox using the two equation differential model of Saffman. In each of these studies it was essential that the full Navier- Stokes equations be considered to describe the viscous- inviscid interaction and the elliptic nature of separating-

A complete model of turbulence

Abstract: A set of constitutive equations suitable for a priori computation of turbulent shear flows has been developed. Since no properties of a given turbulent flow need be known in advance in order to obtain a solution, the equations comprise a complete model of turbulence. Perturbation analysis shows that the model predicts a composite five-layer structure for an incompressible turbulent boundary layer, viz, a defect layer, a law-of-the-wall layer, a viscous sublayer, a near-surface roughness layer, and a viscous superlayer at the boundary-layer edge. Analysis of the defect layer demonstrates the key improvement of the model over its predecessor, the Saffman-Wilcox two-equation model of turbulence. Examination of model-predicted sublayer structure yields model-parameter boundary conditions appropriate for surfaces with roughness and mass injection. Results of numerical computations of compressible and incompressible equilibrium boundary layers show that, for such flows, the model is as accurate as mixing-length theory. Applications to transitional boundary layers and to nonequilibrium relaxation of a boundary layer passing from a rough to a smooth surface indicate that the model's applicability extends far beyond that of mixing-length theory's.

Flow pattern velocity and turbulence energy measurements and predictions in a water model of an argon-stirred ladle

Abstract: Experiments were carried out using a simplified water model of an argon-stirred ladle system. The flow patterns were determined by a flow visualization technique and the velocity and turbulence energy fields were quantitatively measured using hot-film anemometry. The latter quantities were predicted by solving the turbulent Navier-Stokes equations using Spalding’sk-W model for the turbulence viscosity. There is semiquantitative agreement between predictions and measurements. Mixing lengths also were computed. This agreement between measurements and predictions provides further evidence that modeling is a promising approach for the study of recirculating turbulent flows in steel processing operations.

Comments on the diffusion model of turbulent mixing.

Abstract: Several recent investigations of post--main-sequence stellar evolution have used a diffusion model of turbulent mixing, and each author has used a different variation. We justify the diffusion model in the framework of a statistical theory of turbulence and show that the difference between the two diffusion equations that have been proposed is related to an ambiguity in the definition of a Lagrangian hydrodynamical calculation by finite differences in the presence of turbulence. We present difference equations for solving the diffusion equation which theory and experience indicate are free of numerical difficulties. (AIP)

The structure of the turbulence in transient pipe flows

Abstract: The dynamic structure of turbulence in transient pipe flows was studied experimentally. Two types of stepwise change in the flow rate of a fully developed pipe flow showed apparently different behaviours in this turbulence. With a stepwise increase in the flow rate, the dominant feature was the generation and propagation of a new turbulence. With a stepwise decrease in the flow rate the dominant feature was the decay of an old turbulence. However, a comparison of these two types of stepwise change indicates a coherent structure in the propagation of a new turbulence in both transient pipe flows. The propagation time of new turbulence is determined by the condition at its generation. Moreover, this coherent character is also applicable to the propagation of an old turbulence, and the beginning of the decay of the old turbulence is predicted by the propagation time in initial steady state. On the basis of these facts the dynamic behaviours of turbulence in both these types of transient flow are interpreted consistently. For the decay of the old turbulence, the ''linear'' decay law is applicable and the decay rate is governed by the flow condition during decay, although the propagation time is not affected by the transient flow.

Response of a turbulent boundary layer to random fluctuations in the external stream

Abstract: The growth of the turbulent boundary layer is investigated primarily by means of conditional averages in the outer region (i.e., averages inside and outside the bulges limiting the free edge). The main quantities analyzed are: mean velocities, mean velocity gradients, turbulence intensities, Reynolds stresses, kinetic energy balances, skewness and flatness factors of the velocity and Reynolds stress fluctuations, intermittency factor, and indentations of the free edge. Considerable attention is paid to the cumulative straining process of the turbulence by the mean velocity gradient.

Turbulence measurements in hypersonic shock-wave boundary-layer interaction flows

Abstract: Turbulent intensity and Reynolds shear stress measurements are presented for two nonadiabatic hypersonic shock-wave boundary-layer interaction flows, one with and one without separation. These measurements were obtained using a new hot-wire probe specially designed for heated flows. Comparison of the separated and attached flows shows a significant increase above equilibrium values in the turbulent intensity and shear stress downstream of the interaction region for the attached case, while for the separated case, the turbulent fluxes remain close to equilibrium values. This effect results in substantial differences in turbulence lifetime for the two flows. We propose that these differences are due to a coupling between the turbulent energy and separation bubble unsteadiness, a hypothesis supported by the statistical properties of the turbulent fluctuations.

A one-equation model of turbulence for use with the compressible Navier-Stokes equations

Abstract: The glushko one equation model of turbulence is extended to compressible flows without boundary layer approximations.

Some Results from a Simplified Three-Dimensional Numerical Model of Atmospheric Turbulence

Abstract: A simplified set of subgrid-scale transport equations is used to compute the stresses in a three-dimensional model of thermal convection in the atmosphere. Terms appearing in the full transport equations thought not to be essential to the large-scale dynamics are discarded, leaving prognostic equations to be solved for the subgrid-scale energy and the virtual potential temperature variance only. Equations for the Reynolds stresses and the subgrid-scale temperature-velocity correlations are considerably simplified and can be solved algebraically. A scale analysis of the full transport equations is offered as partial justification for the present approach in the case of nearly isotropic turbulence. The problem studied is that of a well-mixed layer bounded above by a region of strong stable stratification. The present model gives a significant improvement in the representation of the large-scale variables as compared with the more conventional eddy viscosity approach. In three experiments testing di...

Constants of motion in models of two‐dimensional turbulence

Abstract: It is shown by four examples, that the truncated spectral model for two‐dimensional flows can have other constants of motion than energy and enstrophy. Thus, for these models the flow cannot be mixing.

Ion-acoustic heating from renormalized turbulence theory

Abstract: Infinite-order perturbation theories of Vlasov turbulence are developed to obtain a renormalized formulation of the mode coupling and quasilinear equations for current-driven ion-acoustic turbulence. The renormalized theory eliminates divergence problems from nonlinear electron interactions and predicts an ion acceleration mechanism producing substantial ion tails at t..omega../subp//subi/ approx. 50 from initially cold-ion distributions. Time-dependent numerical solutions of the turbulence equations are presented, and comparisons are drawn from experiments and computer simulation. (AIP)