A Generalized Integral Computation Technique for Nonequilibrium Compressible Turbulent Boundary Layers Using Moment Equations

1972 ◽  
Vol 39 (3) ◽  
pp. 667-672 ◽  
Author(s):  
J. P. Lamb ◽  
L. J. Hesler ◽  
J. H. Smith
Keyword(s):  
Boundary Layers ◽  
Mechanical Energy ◽  
Moment Equation ◽  
Turbulent Boundary Layers ◽  
Moment Equations ◽  
Governing Equations ◽  
Starting Conditions ◽  
The Moment ◽  
Momentum Integral ◽  
Layer Concept

Computation of nonequilibrium compressible turbulent boundary layers using Coles’ three-parameter representation for the layer (cf, δ, Π) is discussed. Governing equations include momentum integral, skin friction, and an integral moment equation. It is shown that the hypothetical equilibrium layer concept employed by Alber to determine the dissipation integral of the mechanical energy equation can be utilized to estimate similar auxiliary parameters in the entrainment and moment-of-momentum integral equations. A series of comparisons of experimental data and predictions, using each of the moment equations shows that all combinations yield very similar results which are in general agreement with measurements. Some sensitivity to starting conditions was observed with the moment-of-momentum and entrainment relations.

The Departure from Equilibrium of Turbulent Boundary Layers

1968 ◽  
Vol 19 (1) ◽  
pp. 1-19 ◽  
Author(s):  
H. McDonald
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Kinetic Energy ◽  
Stress Distribution ◽  
Boundary Layers ◽  
Turbulent Boundary Layers ◽  
Two Dimensional ◽  
Pressure Distributions ◽  
Momentum Integral ◽  
State Examination

SummaryRecently two authors, Nash and Goldberg, have suggested, intuitively, that the rate at which the shear stress distribution in an incompressible, two-dimensional, turbulent boundary layer would return to its equilibrium value is directly proportional to the extent of the departure from the equilibrium state. Examination of the behaviour of the integral properties of the boundary layer supports this hypothesis. In the present paper a relationship similar to the suggestion of Nash and Goldberg is derived from the local balance of the kinetic energy of the turbulence. Coupling this simple derived relationship to the boundary layer momentum and moment-of-momentum integral equations results in quite accurate predictions of the behaviour of non-equilibrium turbulent boundary layers in arbitrary adverse (given) pressure distributions.

Evaluation of moment equation in the 2000 Canadian highway bridge design code for soil–metal arch structures

2004 ◽  
Vol 31 (2) ◽  
pp. 281-291 ◽  
Author(s):  
Dong-Ho Choi ◽  
Gi-Nam Kim ◽  
Peter M Byrne
Keyword(s):  
Finite Element ◽  
Moment Equation ◽  
Finite Element Analyses ◽  
Moment Equations ◽  
Metal Structures ◽  
Maximum Moment ◽  
Design Variables ◽  
Arch Structures ◽  
The Moment ◽  
Soil Metal

This paper evaluates the moment equation in the 2000 Canadian highway bridge design code (CHBDC) for soil–metal arch structures. This equation is adopted from Duncan's moment equation (1978), which is based on his finding from finite element analyses that the maximum moment occurs at the quarter point of soil-metal structures. However, finite element analyses carried out for this study demonstrate that the maximum moment in soil–metal arch structures with spans greater than approximately 11 m occurs at the crown point. In this study, the location and magnitude of the maximum moment was examined for soil–metal arch structures having spans of 6–20 m under three construction stages; backfill up to the crown, backfill up to the cover depth, and live loading. Based on the location of the maximum moment, two sets of moment equations dependant on span length were found necessary. Moment coefficients and moment reduction factors in moment equations are proposed from the results of numerous finite element analyses for semi-circular arch and part-arch types of soil–metal structures considering the various design variables, such as span length, structural shapes, section properties, and backfill conditions. The validity of the coefficients and reduction factors in the moment equation of the 2000 CHBDC is investigated by comparison with those proposed in this study. The comparison demonstrates that the moment equation of the 2000 CHBDC is still valid and a little conservative. The effects of design variables on the variations of moments of soil–metal arch structures during construction stages are also examined.Key words: soil–metal arch structures, moment equations, CHBDC, soil-structure interaction.

On the Shear-Stress Integral of Turbulent Boundary Layers

1986 ◽  
Author(s):  
H. Pfeil ◽  
M. Göing
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Integral Method ◽  
Turbulent Boundary Layers ◽  
Turbulent Shear ◽  
Two Dimensional ◽  
Basic Assumptions ◽  
Turbulent Shear Stress ◽  
The Moment ◽  
Momentum Integral

The paper presents an integral method to predict turbulent boundary layer behaviour in two-dimensional, incompressible flow. The method is based on the momentum and moment-of-momentum integral equations and a friction law. By means of the compiled data of the 1968-Stanford-Conference, the results show that the integral of the turbulent shear-stress across the boundary layer, which appears in the moment-of-momentum integral equation, can be described by only two basic assumptions for all cases of flow.

Calculation of the Development of Turbulent Boundary Layers With a Turbulent Freestream

1974 ◽  
Vol 96 (4) ◽  
pp. 348-352 ◽  
Author(s):  
R. L. Evans ◽  
J. H. Horlock
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Model Equation ◽  
Plate Boundary ◽  
Adverse Pressure Gradient ◽  
Turbulent Boundary Layers ◽  
Integral Boundary ◽  
Momentum Integral ◽  
Flat Plate Boundary Layer ◽  
Good Agreement

An existing integral boundary layer calculation procedure is modified to predict turbulent boundary layers developing in a turbulent freestream. Extra terms in both the turbulence model equation and the momentum integral equation are introduced to account for the effects of freestream turbulence. Good agreement with flat plate boundary layer measurements in a turbulent freestream has been obtained, while comparison with measurements in a severe adverse pressure gradient shows qualitative agreement with experiments.

A Simple Integral Method for the Calculation of Thick Axisymmetric Turbulent Boundary Layers

1974 ◽  
Vol 25 (1) ◽  
pp. 47-58 ◽  
Author(s):  
V C Patel
Keyword(s):  
Boundary Layer ◽  
Integral Equation ◽  
Turbulent Boundary Layer ◽  
Velocity Profile ◽  
Boundary Layers ◽  
Integral Method ◽  
Turbulent Boundary Layers ◽  
Acceptable Accuracy ◽  
Approximate Form ◽  
Momentum Integral

SummaryA simple integral method is described for the calculation of a thick axisymmetric turbulent boundary layer. It is shown that the development of the boundary layer can be predicted with acceptable accuracy by using an approximate form of the momentum-integral equation, an appropriate skin-friction law, and an entrainment equation obtained for axisymmetric boundary layers. The method also involves the explicit use of a velocity profile family in order to interrelate some of the integral parameters. Available experimental results have been used to demonstrate the general accuracy of the method.

On the Shear-Stress Integral of Turbulent Boundary Layers

1987 ◽  
Vol 109 (3) ◽  
pp. 398-404
Author(s):  
H. Pfeil ◽  
M. Go¨ing
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Integral Method ◽  
Turbulent Boundary Layers ◽  
Turbulent Shear ◽  
Basic Assumptions ◽  
Turbulent Shear Stress ◽  
Layer Behavior ◽  
The Moment ◽  
Momentum Integral

The paper presents an integral method to predict turbulent boundary layer behavior in two-dimensional, incompressible flow. The method is based on the momentum and moment-of-momentum integral equations and a friction law. By means of the compiled data of the 1968 Stanford Conference, the results show that the integral of the turbulent shear stress across the boundary layer, which appears in the moment-of-momentum integral equation, can be described using only two basic assumptions for all cases of flow.

Prediction of the Development of Non-Equilibrium Turbulent Boundary Layers Based on Modified Log-Wake Law

2014 ◽  
Vol 620 ◽  
pp. 39-43
Author(s):  
Tao Zhang ◽  
Xiao Jun Zhu ◽  
Fei Peng ◽  
Shao Song Min
Keyword(s):  
Dynamical System ◽  
Boundary Layers ◽  
System Approach ◽  
Reasonable Agreement ◽  
Nonlinear Dynamical System ◽  
Turbulent Boundary Layers ◽  
Experiment Data ◽  
Nonlinear Dynamical ◽  
Momentum Integral ◽  
Non Equilibrium

The properties of non-equilibrium turbulent boundary layers are substantially more complicated than that of equilibrium ones, and current understanding and predictive capabilities of the former are less well developed than of the latter. This paper proposed a nonlinear dynamical system approach to predict streamwise development of non-equilibrium turbulent boundary layers by means of realizing the closure of the momentum integral equation with aid of the modified log-wake law and the entrainment equation. The example calculation showed the results were in reasonable agreement with the experiment data, and demonstrated the proposed method could predict the streamwise evolution of the layers accurately and simply. Moreover, the method would be conveniently extended to the flows over rough surfaces.

scholarly journals Compartor: a toolbox for the automatic generation of moment equations for dynamic compartment populations

2021 ◽  
Author(s):  
Tobias Pietzsch ◽  
Lorenzo Duso ◽  
Christoph Zechner
Keyword(s):  
Stochastic Dynamics ◽  
Moment Equation ◽  
Automatic Generation ◽  
Moment Equations ◽  
Biochemical System ◽  
Biochemical Processes ◽  
Stochastic Population Models ◽  
Living Organisms ◽  
The Moment ◽  
Python Package

Abstract Summary Many biochemical processes in living organisms take place inside compartments that can interact with each other and remodel over time. In a recent work, we have shown how the stochastic dynamics of a compartmentalized biochemical system can be effectively studied using moment equations. With this technique, the time evolution of a compartment population is summarized using a finite number of ordinary differential equations, which can be analyzed very efficiently. However, the derivation of moment equations by hand can become time-consuming for systems comprising multiple reactants and interactions. Here we present Compartor, a toolbox that automatically generates the moment equations associated with a user-defined compartmentalized system. Through the moment equation method, Compartor renders the analysis of stochastic population models accessible to a broader scientific community. Availability and implementation Compartor is provided as a Python package and is available at https://pypi.org/project/compartor/. Source code and usage tutorials for Compartor are available at https://github.com/zechnerlab/Compartor.

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