Liquid Acquisition Devices for Advanced In-Space Cryogenic Propulsion Systems

The special importance of turbulence maybe comprehended by its ubiquity in innumerable natural and technical systems. Examples for natural turbulent flows are the atmospheric flow and the oceanic current which to calculate is a crucial point in climate research. Classical engineering application involving turbulence are the flows around airplanes or cars or turbulence within jet or reciprocating engines. Though supposed for more than hundred years, only with the advent of super computers it became apparent that the Navier-Stokes equations provide an excellent continuummechanical model for turbulent flows. Still, the exclusive and direct application of the Navier-Stokes equations to practical flow problems at very high Reynolds numbers without invoking any additional assumptions is still several decades away. However, in most applications it is not at all necessary to know all the detailed fluctuations of velocity and pressure present in turbulent flows but for the most part statistical measures are sufficient. This was in fact the key idea of O. Reynolds who was the first to suggest a statistical description of turbulence. The Navier-Stokes equations, however, constitute a non-linear and, due to the pressure Poisson equation, a non-local set of equations. As an immediate consequence of this the equations for the mean or expectation values for velocity and pressure lead to an infinite set of statistical equations, or, if truncated at some level of statistics, an un-closed system is generated.

Symmetries and their importance for statistical turbulence theory

An incompressible, pressure-driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary-value problem for a second-order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. There are three characteristic flow regions in the channel: the core region and two wall regions near injection and suction walls. For each region, the solution is constructed. The asymptotic matching gives formulas for the wall shear stress and the maximum mean velocity. A limit transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one-parameter family of curves. The theoretical results obtained are in good agreement with available direct numerical simulation data.

Analytical study of turbulent Poiseuille flow with wall transpiration

A turbulent channel flow facility was used to study the scaling of the combined effects of roughness and flow injection on the mean flow and turbulence characteristics of turbulent plane Poiseuille flow. It was found that the additional momentum injection through the surface enhanced the roughness effects and, for the mean flow, the effect of blowing was indistinguishable from that of increased roughness. This analogy broke down for the turbulence statistics in that the addition of blowing resulted in behavior which did not follow that predicted by Townsend’s hypothesis. Instead, the outer-scaled Reynolds stress was found to deviate from that for the rough-walled boundary condition without blowing well into the outer layer. It was found that this deviation from the expected Reynolds stress scaling behavior was caused by the suppression of kinetic energy content associated with large-scale motions.

Investigation of the scaling of roughness and blowing effects on turbulent channel flow

In the present work, scaling laws for special turbulent flow phenomena are investigated using a mathematical method based on Lie-point symmetries. Moreover, the theoretical results are compared to available DNS data.

At first, a set of governing partial differential equations (PDEs), here the multi-point correlation equations, is introduced, describing a turbulent flow of a Newtonian fluid with constant density. Lie-point symmetries, which represent transformations of functions and variables leaving the form of a differential equation unchanged, are determined for this set of differential equations. After stating general conditions for these symmetries, the classical symmetries originating from the Navier-Stokes equations are derived. Then, it is shown that even more symmetries exist for the multi-point correlation equations.

If the symmetries are known, a mathematical algorithm can be applied in order to derive special solutions for the mean velocity and the components of the Reynolds stress tensor, which represent the desired scaling laws. This procedure is performed for homogeneous turbulence, channel flows at the near-wall region and the core region, boundary layers, channel flows with transpiration and rotating channel flows, where the rotational axis can lie in the spanwise or the wall-normal direction. The results are compared to solutions obtained by other theoretical approaches as well as to available numerical data. For all studied cases, a convincing agreement with the data can be obtained.

Scaling Laws in Turbulence - A Theoretical Approach Using Lie-Point Symmetries

Steady turbulent viscous incompressible fluid flow in a plane channel is calculated for the case of uniform blowing and suction through opposite walls. There are no experimental data for flows of this type. The flows were calculated by two methods: a direct numerical simulation method and using a three-parameter turbulence model. Direct numerical simulation was carried out using the same (apart from the boundary conditions) algorithm for numerical solution of the Navier-Stokes equations as that used earlier for calculating flows in pipes and channels with impermeable walls. In the second group of calculations the version of the model published in 1978 was used. The results obtained by the two methods are in good agreement. The difference is within the spread of the experimental data used for determining the parameters of the model. The agreement obtained makes it possible to assert that the turbulence direct numerical simulation algorithm developed can be used for the analysis of flows with quite different boundary conditions, including cases where there are no corresponding experimental data.

Turbulent flow in a channel with permeable walls. Direct numerical simulation and results of three-parameter model

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Turbulent flow in a channel with permeable walls. Direct numerical simulation and results of three-parameter model