Engineering topics
Publications and source records attributed to Deissler, R. G..
Whether turbulent solutions of the Navier-Stokes equations are chaotic is considered. Initially neighboring solutions for a low-Reynolds-number fully developed turbulence are compared. The turbulence is sustained by a nonrandom time-independent external force. The solutions separate exponentially with time, having a positive Liapunov characteristic exponent. Thus the turbulence is characterized as chaotic.
Analytical and numerical approaches to the mechanics of turbulent flow are examined in a general introduction and illustrated with graphs. Topics discussed include the averaged and unaveraged basic equations, numerical solutions and methods, homogeneous fluctuations and turbulence with no mean flow, uniformly sheared fluctuations and turbulence, inhomogeneous fluctuations and turbulence in a developing shear layer, and steady-state homogeneous turbulence with a spatially periodic body force.
Some turbulent solutions of the unaveraged Navier-Stokes equations (equations of fluid motion) are reviewed. Those equations are solved numerically in order to study the nonlinear physics of incompressible turbulent flow. The three components of the mean-square velocity fluctuations are initially equal for the conditions chosen. The resulting solutions show characteristics of turbulence, such as the linear and nonlinear excitation of small-scale fluctuations. For the stronger fluctuations the initially nonrandom flow develops into an apparently random turbulence. The cases considered include turbulence that is statistically homogeneous or inhomogeneous and isotropic or anisotropic. A statistically steady-state turbulence is obtained by using a spatially periodic body force. Various turbulence processes, including the transfer of energy between eddy sizes and between directional components and the production, dissipation, and spatial diffusion of turbulence, are considered. It is concluded that the physical processes occurring in turbulence can be profitably studied numerically.
Some turbulent solutions of the unaveraged Navier-Stokes equations (equations of fluid motion) are reviewed. Those equations are solved numerically in order to study the nonlinear physics of incompressible turbulent flow. Initial three-dimensional cosine velocity fluctuations and periodic boundary conditions are used in most of the work considered. The three components of the mean-square velocity fluctuations are initially equal for the conditions chosen. The resulting solutions show characteristics of turbulence such as the linear and nonlinear excitation of small-scale fluctuations. For the stronger fluctuations, the initially nonrandom flow develops into an apparently random turbulence. Thus randomness or turbulence can arise as a consequence of the structure of the Navier-Stokes equations. The cases considered include turbulence which is statistically homogeneous or inhomogeneous and isotropic or anisotropic. A mean shear is present in some cases. A statistically steady-state turbulence is obtained by using a spatially periodic body force. Various turbulence processes, including the transfer of energy between eddy sizes and between directional components, and the production, dissipation, and spatial diffusion of turbulence, are considered. It is concluded that the physical processes occurring in turbulence can be profitably studied numerically.
To study the nonlinear physics of inhomogeneous turbulent shear flow, the unaveraged Navier-Stokes equations are solved numerically. For initial conditions a three-dimensional cosine velocity fluctuation and a mean-velocity profile with a step are used. Although the initial conditions are nonrandom. The flow soon becomes turbulent. Concentrated turbulent energy develops near the plane where the mean velocity gradient is initially infinite. The terms in the one-point correlation equation for turbulent energy, including those for the diffusion and production of turbulence, are calculated, the diffusion terms tend to make the turbulence more homogeneous.
It is noted that several terms in the two-point spectral equation for homogeneous turbulence can be interpreted as spectral-transfer terms; that is, they represent the net rate of energy transfer into a wavenumber region from all other wavenumbers. This holds for terms associated with both turbulence and self-interaction and interaction between turbulence and mean gradients. It is not seen as obvious, however, that similar interpretations apply when the turbulence is not homogeneous. In particular, one might question the interpretation for the terms associated with turbulence self-interaction because the condition of homegeneity is generally used in making the interpretation. It is the purpose here to consider whether terms interpretable as transfer terms exist in the equations for inhomogeneous turbulence. It is found that certain terms in the two-point spectral equation can be interpreted as transfer terms.
To study the nonlinear physics of uniform turbulent shear flow, the unaveraged Navier-Stokes equations are solved numerically. This extends our previous work in which mean gradients were absent. For initial conditions, modified three-dimensional-cosine velocity fluctuations are used. The boundary conditions are modified periodic conditions on a stationary three-dimensional numerical grid. A uniform mean shear is superimposed on the initial and boundary conditions. The three components of the mean-square velocity fluctuations are initially equal for the conditions chosen. As in the case of no shear the initially nonrandom flow develops into an apparently random turbulence at higher Reynolds number. Thus, randomness or turbulence can apparently arise as a consequence of the structure of the Navier-Stokes equations. Except for an initial period of adjustment, all fluctuating components grow with time. The initial equality of the three intensity components is destroyed by the shear, the transverse components becoming smaller than the longitudinal one, in agreement with experiment. Also, the shear creates a small-scale structure in the turbulence. The nonlinear solutions are compared with linearized ones.
To study the nonlinear physics of incompressible turbulent flow, the unaveraged Navier-Stokes equations are solved numerically. Initial three-dimensional cosine velocity fluctuations and periodic boundary conditions are used. No mean gradients are present. The three components of the mean-square velocity fluctuations are equal for the initial conditions chosen. The resulting solution shows characteristics of turbulence, such as the nonlinear excitation of small-scale fluctuations. For the higher Reynolds numbers the initially nonrandom flow develops into an apparently random turbulence.
The unaveraged Navier-Stokes equations are solved numerically in order to study the nonlinear physics of incompressible turbulent flow. Initial three dimensional cosine velocity fluctuations and periodic boundary conditions are used. No mean gradients are present. The three components of the mean square velocity fluctuations are equal for the initial conditions chosen. The resulting solution shows characteristics of turbulence, such as the nonlinear excitation of small scale fluctuations. For the higher Reynolds numbers the initially nonrandom flow develops into an apparently random turbulence.
Turbulence decay is calculated by using experimental initial conditions and discarding quadruple-correlation terms in the correlation equations. Agreement with experiment is good only for moderately small times, but there are no perceptible negative spectral energies even at large times.
Evolution of a rotating flow in a body of fluid bounded by a stationary flat surface is discussed. The calculated results show that the radial pressure gradient is substantially reduced in the region close to the surface, so that letting that gradient be independent of distance from the surface would be expected to give only rough or qualitative estimates. However, the reduced rotation near the stationary surface is still large enough to cause an inflow near the surface and to set up a recirculation pattern. The concentration of vorticity by the radial inflow is not great enough to increase the tangential velocities near the center of rotation.
The turbulence equations are closed by specification of initial conditions (using either a Taylor or an exponential series) and by a modified Kovasznay-type closure. Good results for large times are obtained only for the initial-conditions closure used with four or more terms of an exponential series. The evolution of all of the initially-specified spectra can be calculated rather well from the theory. From a fundamental standpoint the method thus seems to be satisfactory.
The turbulence equations are closed by specification of initial conditions (using either a Taylor or an exponential series) and by a modified Kovasznay-type closure. Good results for large times are obtained only for the initial-conditions closure used with four or more terms of an exponential series. The evolution of all of the initially-specified spectra can be calculated rather well from the theory. From a fundamental standpoint the method thus seems to be satisfactory.
Data for the energy transfer function are used to estimate the degree of localness of energy transfer in homogeneous turbulence. It is found that in regions where the energy which enters a wavenumber band is greater than the energy leaving, much of the energy entering the band is produced by wavenumbers an order of magnitude smaller. Thus for both low and high Reynolds numbers, spectral energy transfer is nonlocal. The tendency of the energy to jump between separated wavenumber regions agrees with the theory that turbulence forms concentrated regions of large velocity gradients. It is also felt that the universal equilibrium theory may be applicable if the Reynolds number of the turbulence is very high.
A frictionless adiabatic model is used to study the growth of random vortices in an atmosphere with buoyant instability and vertical wind shear, taking account of the effects of axial drag, heat transfer and precipitation-induced downdrafts. It is found that downdrafts of tornadic magnitude may occur in negatively buoyant columns. The radial-inflow velocity required to maintain a given maximum tangential velocity in a tornado is determined by using a turbulent vortex model. A tornado model which involves a rotating parent cloud as well as buoyancy and precipitation effects is also discussed.
The problem of closure in turbulence in the case of two-point correlations resides in the existence of two unknowns E and W, the energy spectrum function and the transfer function, respectively, in the spectrum equation. In the case of weak turbulence, W is negligible. In case of higher correlations, closure can be effective by neglecting the inertia term in the highest order term used. Specifying a certain number of spectra at an initial time is also a way of getting around the closure problem. A simple case of turbulent shear flow is then considered, where two-point correlation equations are used and the velocity is broken into mean and fluctuating components. This yields a differential equation for the energy spectrum, the three terms of which are the energy spectrum, production term and dissipation term. They are plotted for a particular time. Similar analyses and comparisons with experiment are made for pipe and boundary layer flows.
The proposed approach to the derivation of the Navier-Stokes equation is thought to be more plausible and easier to understand than other derivations that can be found in works on fluid mechanics. The tensor character of the stress is central to the derivation. In particular, a linear relation between stress and strain rate is assumed only for the shear, rather than for the full stress tensor as is done in most other derivations. An assumption for the shear is naturally simpler and easier to verify experimentally. The use of tensor analysis is shown to greatly simplify the derivation.
The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitational and rotational terms in the equations are of first order in the space variables, the pressure-gradient terms are of second order, and the turbulent-viscosity term is of third order. The presence of turbulent viscosity ensures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial flow is always inward and allows collapse to occur (axially) even when the rotation is large. An approximate solution of the governing partial differential equations is also given in order to study the spatial distributions of the density and velocity.