Joan G. Moore

Realizability in two-equation turbulence models

For calculated Reynolds normal stresses to be physically realizable they should be positive in all directions and bounded by 2k. The commonly used twoequation turbulence models give realizable stresses for simple boundary layers but can give non-realizable stresses for geometries where pressure gradients impose an inviscid velocity strain field for example, flow approaching the leading edge of a body or impinging on a flat plat. The erroneous calculated stresses far from the walls can cause severe over-production of turbulence kinetic energy which in turn can seriously compromise calculated boundary layer development. This paper reviews some methods and models which ensure realizability in two-equation Boussinesq and algebraic Reynolds stress models.

Entropy Production Rates From Viscous Flow Calculations: Part I — A Turbulent Boundary Layer Flow

A procedure for obtaining entropy production rates from viscous flow calculations is described. The method is based on process thermodynamics; it allows loss production to be calculated in “irreversible equilibrium processes.”The two-dimensional turbulent boundary layer of Samuel and Joubert is considered. Mean rates of entropy production are evaluated from measured data using rates of dissipation and rates of increase of turbulence kinetic energy. Calculations performed with the Moore Cascade Flow Program give good agreement with mean rates of entropy production and reveal details of the distribution of entropy production throughout the boundary layer. This method is used in Part II to reveal loss sources in calculations for a rectangular elbow.Copyright

Entropy Production Rates From Viscous Flow Calculations: Part II — Flow in a Rectangular Elbow

Entropy creation is studied in a three-dimensional turbulent flow in a rectangular elbow with 90 degrees of turning. In Part I, a method of evaluating entropy production rates in calculations of viscous flow was described. This method is used here to show where the calculated losses arise in the flow field.The flow develops strong passage vortices as large shear layers at the inlet develop a streamwise component of vorticity. The entropy production in the shear layers is compared with the entropy creation in the wall boundary layers. The influence of the three-dimensional flow on the distributions of entropy production in the bend is discussed.Copyright

Methods of Classical Mechanics Applied to Turbulence Stresses in a Tip Leakage Vortex

Moore et al. measured the six Reynolds stresses in a tip leakage vortex in a linear turbine cascade. Stress tensor analysis, as used in classical mechanics, has been applied to the measured turbulence stress tensors. Principal directions and principal normal stresses are found. A solid surface model, or 3-d glyph, for the Reynolds stress tensor is proposed and used to view the stresses throughout the tip leakage vortex. Modelled Reynolds stresses using the Boussinesq approximation are obtained from the measured mean velocity strain rate tensor. The comparison of the principal directions and the 3-d graphical representations of the strain and Reynolds stress tensors aids in the understanding of the turbulence and what is required to model it.Copyright

Realizability in Turbulence Modelling for Turbomachinery CFD

It is obvious that the Reynolds normal stresses Display Formulauu¯ should always be positive in all directions, i.e. the computed turbulence stresses should be realizable. However, the commonly used two-equation turbulence models do not incorporate realizability. They take the turbulent viscosity as cμk2/e with cμ a constant, and frequently generate negative normal stresses far from walls in the nominally inviscid sections of turbomachinery flows. Pressure gradients due to leading edge stagnation and blade turning create an inviscid strain field. These strains cause the calculation of negative normal stresses over significant portions of the flow field. The result can be erroneous increases in turbulence kinetic energy upstream of the leading edge by a factor of ten or more. This erroneous turbulence is then convected around the blade and through the blade row, significantly affecting the computed boundary layer development and profile losses.Frequently the problem of overproduction is avoided by using artificially high values of the dissipation, e, at the inlet. But this incorrect procedure is not needed when realizability is incorporated in the turbulence model.The paper reviews some methods and models which ensure realizability in two-equation turbulence models. The extent of the problem and its solution are illustrated with examples from compressor and turbine cascades.Copyright

Calculation of Horseshoe Vortex Flow Without Numerical Mixing

The usefulness of three-dimensional flow calculations has frequently been obscured by the numerical mixing present in the calculation methods. This paper describes a new method of forming the finite difference momentum equations. The new method results in well posed equations which introduce no numerical mixing. It may be used with orthogonal or non-orthogonal grids and with uniform or highly non-uniform grid spacing.The method is demonstrated by comparing it with upwind differencing on the calculation of a simple example. It is then used in an elliptic pressure-correction calculation procedure to calculate a leading edge horseshoe vortex about a Rankine half body. The results compare well with the experimental data presented in a companion paper.Copyright

A Calculation Procedure for Three-Dimensional, Viscous, Compressible Duct Flow. Part II—Stagnation Pressure Losses in a Rectangular Elbow

Three-dimensional, turbulent flow is calculated in an elbow used by Stanitz for an experimental investigation of secondary flow. Calculated wall-static pressure distributions and distributions of stagnation-pressure loss, both spatial and as a function of mass-flow ratio, are in good agreement with Stanitz’ measurements, justifying the use of a relatively simple mixing-length viscosity model. The calculation procedure and the results of two-dimensional “inviscid” flow calculations used as the starting point for the present calculations are described in Part I of this paper. The computed flow field shows clearly the development of the passage vortices.

Secondary Flow, Separation and Losses in the NACA 48-Inch Centrifugal Impeller at Design and Off-Design Conditions

An elliptic flow calculation procedure has been used to model 3-D flow in the NACA 48–inch centrifugal impeller. The results demonstrate that fully elliptic steady flow calculations can be performed at design and off-design conditions. The calculations reproduce the measured overall performance and most of the features of the loss distributions observed in the NACA flow study. They give further insight into the complex 3-D flow with leading-edge separation and tip leakage.The calculated secondary flow patterns are presented and used to explain the convection of vortices in a more recent laser anemometry study of a centrifugal compressor impeller.Copyright

New Twists on the Deverson Axial-Flow Compressor Rotor

Goto has presented data in the literature for the Deverson axial flow compressor rotor in a single stage environment. The flow features tip leakage and hub corner stall. In this paper, 3-d turbulent steady flow calculations for the rotor are used to determine if, and how, this rotor can be a useful CFD test case. The sensitivity of the CFD results to unknowns about the test case environment is computed. In particular the influences of blade twist, hub gap leakage flow between rotating and stationary components, and a departure from a radial equilibrium pressure distribution at the rotor exit are investigated. Evidence in support of these phenomena is presented and discussed, their likely magnitudes are suggested, and their influence on the rotor performance is quantified.Copyright

Osborne Reynolds: Energy Methods in Transition and Loss Production — A Centennial Perspective

Osborne Reynolds’ developments of the concepts of Reynolds averaging, turbulence stresses, and equations for mean kinetic energy and turbulence energy are viewed in the light of one-hundred years of subsequent flow research. Attempts to use Reynolds energy-balance method to calculate the lower critical Reynolds number for pipe and channel flows is reviewed. The modern use of turbulence-energy methods for boundary layer transition modelling is discussed, and a current European Working Group effort to evaluate and develop such methods is described. The possibility of applying these methods to calculate transition in pipe, channel and sink flows is demonstrated using a one-equation, q-L, turbulence model. Recent work using the equation for the kinetic energy of mean motion to gain understanding of loss production mechanisms in three-dimensional turbulent flows is also discussed.Copyright