# Turbulent Boundary Layers & Turbulence Models. Lecture 09

Size: px
Start display at page:

Transcription

1 Turbulent Boundary Layers & Turbulence Models Lecture 09

2 The turbulent boundary layer In turbulent flow, the boundary layer is defined as the thin region on the surface of a body in which viscous effects are important. The boundary layer allows the fluid to transition from the free stream velocity U to a velocity of zero at the wall. The velocity component normal to the surface is much smaller than the velocity parallel to the surface: v << u. The gradients of the flow across the layer are much greater than the gradients in the flow direction. The boundary layer thickness δ is defined as the distance away from the surface where the velocity reaches 99% of the free-stream velocity. δ = y, where u U = 0.99

3 Drag on a smooth circular cylinder The drag coefficient is defined as follows: no separation steady separation unsteady vortex shedding laminar BL wide turbulent wake turbulent BL narrow turbulent wake Turbulent flow reduced drag

4 The turbulent boundary layer

5 The turbulent boundary layer Important variables: Distance from the wall: y. Wall shear stress: τ w. The force exerted on a flat plate is the area times the wall shear stress. Density: ρ. Dynamic viscosity: µ. Kinematic viscosity: ν. Velocity at y: U. The friction velocity: u τ = (τ w /ρ) 1/2. We can define a Reynolds number based on the distance to the wall using the friction velocity: y + = yu τ /ν. We can also make the velocity at y dimensionless using the friction velocity: u + = U/ u τ.

6 Boundary layer structure u+=y+ y+=1

7 Standard wall functions The experimental boundary layer profile can be used to calculate τ w. However, this requires y+ for the cell adjacent to the wall to be calculated iteratively. In order to save calculation time, the following explicit set of correlations is usually solved instead: Here: U p is the velocity in the center of the cell adjacent to the wall. y p is the distance between the wall and the cell center. k p is the turbulent kinetic energy in the cell center. κ is the von Karman constant (0.42). E is an empirical constant that depends on the roughness of the walls (9.8 for smooth surfaces).

8 Near-wall treatment - momentum equations The objective is to take the effects of the boundary layer correctly into account without having to use a mesh that is so fine that the flow pattern in the layer can be calculated explicitly. Using the no-slip boundary condition at wall, velocities at the nodes at the wall equal those of the wall. The shear stress in the cell adjacent to the wall is calculated using the correlations shown in the previous slide. This allows the first grid point to be placed away from the wall, typically at 50 < y + < 500, and the flow in the viscous sublayer and buffer layer does not have to be resolved. This approach is called the standard wall function approach. The correlations shown in the previous slide are for steady state ( equilibrium ) flow conditions. Improvements, non-equilibrium wall functions, are available that can give improved predictions for flows with strong separation and large adverse pressure gradients.

9 Two-layer zonal model A disadvantage of the wallfunction approach is that it relies on empirical correlations. The two-layer zonal model does not. It is used for low-re flows or flows with complex near-wall phenomena. Zones distinguished by a walldistance-based turbulent ρ k y Reynolds number: Re y µ Rey >200 The flow pattern in the boundary layer is calculated explicitly. Regular turbulence models are used in the turbulent core region. Only k equation is solved in the viscosity-affected region. ε is computed using a correlation for the turbulent length scale. Zoning is dynamic and solution adaptive. Rey < 200

10 Near-wall treatment - turbulence The turbulence structure in the boundary layer is highly anisotropic. ε and k require special treatment at the walls. Furthermore, special turbulence models are available for the low Reynolds number region in the boundary layer. These are aptly called low Reynolds number models. This is still a very active area of research 35

11 Computational grid guidelines Wall Function Approach Two-Layer Zonal Model Approach First grid point in log-law region: 50 y Gradual expansion in cell size away from the wall. Better to use stretched quad/hex cells for economy. First grid point at y+ 1. At least ten grid points within buffer and sublayers. Better to use stretched quad/hex cells for economy.

12 Obtaining accurate solutions When very accurate (say 2%) drag, lift, or torque predictions are required, the boundary layer and flow separation require accurate modeling. The following practices will improve prediction accuracy: Use boundary layer meshes consisting of quads, hexes, or prisms. Avoid using pyramid or tetrahedral cells immediately adjacent to the wall. After converging the solution, use the surface integral reporting option to check if y+ is in the right range, and if not refine the grid using adaption. For best predictions use the two-layer zonal model and completely resolve the flow in the whole boundary layer. tetrahedral volume mesh is generated automatically prism layer efficiently resolves boundary layer triangular surface mesh on car body is quick and easy to create

13 Summary Boundary layers require special treatment in the CFD model. The influence of pressure gradient on boundary layer attachment showed that an adverse pressure gradient gives rise to flow separation. For accurate drag, lift, and torque predictions, the boundary layer and flow separation need to be modeled accurately. This requires the use of: A suitable grid. A suitable turbulence model. Higher order discretization. Deep convergence using the force to be predicted as a convergence monitor.

14 Turbulence models

15 Turbulence models A turbulence model is a computational procedure to close the system of mean flow equations. For most engineering applications it is unnecessary to resolve the details of the turbulent fluctuations. Turbulence models allow the calculation of the mean flow without first calculating the full time-dependent flow field. We only need to know how turbulence affected the mean flow. In particular we need expressions for the Reynolds stresses. For a turbulence model to be useful it: must have wide applicability, be accurate, simple, and economical to run.

16 Common turbulence models Classical models. Based on Reynolds Averaged Navier-Stokes (RANS) equations (time averaged): 1. Zero equation model: mixing length model. 2. One equation model: Spalart-Almaras. 3. Two equation models: k-ε style models (standard, RNG, realizable), k-ω model, and ASM. 4. Seven equation model: Reynolds stress model. The number of equations denotes the number of additional PDEs that are being solved. Large eddy simulation. Based on space-filtered equations. Time dependent calculations are performed. Large eddies are explicitly calculated. For small eddies, their effect on the flow pattern is taken into account with a subgrid model of which many styles are available.

17 Prediction Methods

18 Boussinesq hypothesis Many turbulence models are based upon the Boussinesq hypothesis. It was experimentally observed that turbulence decays unless there is shear in isothermal incompressible flows. Turbulence was found to increase as the mean rate of deformation increases. Boussinesq proposed in 1877 that the Reynolds stresses could be linked to the mean rate of deformation. Using the suffix notation where i, j, and k denote the x-, y-, and z- directions respectively, viscous stresses are given by: Similarly, link Reynolds stresses to the mean rate of deformation:

19 Turbulent viscosity A new quantity appears: the turbulent viscosity µt. Its unit is the same as that of the molecular viscosity: Pa.s. It is also called the eddy viscosity. We can also define a kinematic turbulent viscosity: ν t = µ t /ρ. Its unit is m 2 /s. The turbulent viscosity is not homogeneous, i.e. it varies in space. It is, however, assumed to be isotropic. It is the same in all directions. This assumption is valid for many flows, but not for all (e.g. flows with strong separation or swirl).

20 Turbulent Schmidt number The turbulent viscosity is used to close the momentum equations. We can use a similar assumption for the turbulent fluctuation terms that appear in the scalar transport equations. For a scalar property φ(t) = Φ + ϕ (t): Here Γt is the turbulent diffusivity. The turbulent diffusivity is calculated from the turbulent viscosity, using a model constant called the turbulent Schmidt number Experiments have shown that the turbulent Schmidt number is nearly constant with typical values between 0.7 and 1.

21 Predicting the turbulent viscosity The following models can be used to predict the turbulent viscosity: Mixing length model. Spalart-Allmaras model. Standard k-ε model. k-ε RNG model. Realizable k-ε model. k-ω model. We will discuss some of these models

22 Mixing length model Further reading On dimensional grounds one can express the kinematic turbulent viscosity as the product of a velocity scale and a length scale: If we then assume that the velocity scale is proportional to the length scale and the gradients in the velocity (shear rate, which has dimension 1/s): we can derive Prandtl s (1925) mixing length model: Algebraic expressions exist for the mixing length for simple 2-D flows, such as pipe and channel flow.

23 Mixing length model discussion Advantages: Easy to implement. Fast calculation times. Good predictions for simple flows where experimental correlations for the mixing length exist. Disadvantages: Completely incapable of describing flows where the turbulent length scale varies: anything with separation or circulation. Only calculates mean flow properties and turbulent shear stress. Use: Sometimes used for simple external aero flows. Pretty much completely ignored in commercial CFD programs today. Much better models are available.

24 Spalart-Allmaras one-equation model Solves a single conservation equation (PDE) for the turbulent viscosity: This conservation equation contains convective and diffusive transport terms, as well as expressions for the production and dissipation of ν t. Developed for use in unstructured codes in the aerospace industry. Economical and accurate for: Attached wall-bounded flows. Flows with mild separation and recirculation. Weak for: Massively separated flows. Free shear flows. Decaying turbulence. Because of its relatively narrow use we will not discuss this model in detail.

25 The k-ε model The k-ε model focuses on the mechanisms that affect the turbulent kinetic energy (per unit mass) k. The instantaneous kinetic energy k(t) of a turbulent flow is the sum of mean kinetic energy K and turbulent kinetic energy k: ε is the dissipation rate of k. If k and ε are known, we can model the turbulent viscosity as: We now need equations for k and ε.

26 Mean flow kinetic energy K The equation for the mean kinetic energy is obtained my multiplying U, V and W to the Reynolds equation and it is as follows: Here Eij is the mean rate of deformation tensor. This equation can be read as: (I) the rate of change of K, plus (II) transport of K by convection, equals (III) transport of K by pressure, plus (IV) transport of K by viscous stresses, plus (V) transport of K by Reynolds stresses, minus (VI) rate of dissipation of K, minus (VII) turbulence production.

27 Turbulent kinetic energy k The equation for the turbulent kinetic energy k is as follows: Here eij is fluctuating component of rate of deformation tensor. This equation can be read as: (I) the rate of change of k, plus (II) transport of k by convection, equals (III) transport of k by pressure, plus (IV) transport of k by viscous stresses, plus (V) transport of k by Reynolds stresses, minus (VI) rate of dissipation of k, plus (VII) turbulence production.

28 Model equation for k The equation for k contains additional turbulent fluctuation terms, that are unknown. Again using the Boussinesq assumption, these fluctuation terms can be linked to the mean flow. The following (simplified) model equation for k is commonly used. The Prandtl number σ k connects the diffusivity of k to the eddy viscosity. Typically a value of 1.0 is used.

29 Turbulent dissipation The equations look quite similar. However, the k equation mainly contains primed quantities, indicating that changes in k are mainly governed by turbulent interactions. Furthermore, term (VII) is equal in both equations. But it is actually negative in the K equation (destruction) and positive in the k equation: energy transfers from the mean flow to the turbulence. The viscous dissipation term (VI) in the k equation 2µeij'.eij' describes the dissipation of k because of the work done by the smallest eddies against the viscous stresses. We can now define the rate of dissipation per unit mass ε as: ε = 2ν eij'.eij'

30 Model equation for ε A model equation for ε is derived by multiplying the k equation by (ε/k) and introducing model constants. The following (simplified) model equation for ε is commonly used. The Prandtl number σε connects the diffusivity of ε to the eddy viscosity. Typically a value of 1.30 is used. Typically values for the model constants C1ε and C2ε of 1.44 and 1.92 are used.

31 k-ε model discussion Advantages: Relatively simple to implement. Leads to stable calculations that converge relatively easily. Reasonable predictions for many flows. Disadvantages: Poor predictions for: swirling and rotating flows, flows with strong separation, axisymmetric jets, certain unconfined flows, and fully developed flows in non-circular ducts. Valid only for fully turbulent flows. Simplistic ε equation.

32 More two-equation models The k-ε model was developed in the early 1970s. Its strengths as well as its shortcomings are well documented. Many attempts have been made to develop two-equation models that improve on the standard k-ε model. We will discuss some here: k-ε RNG model. k-ε realizable model. k-ω model. Algebraic stress model. Non-linear models.

33 Improvement: RNG k- ε k-ε equations are derived from the application of a rigorous statistical technique (Renormalization Group Method) to the instantaneous Navier-Stokes equations. Similar in form to the standard k-ε equations but includes: Additional term in ε equation for interaction between turbulence dissipation and mean shear. The effect of swirl on turbulence. Analytical formula for turbulent Prandtl number. Differential formula for effective viscosity. Improved predictions for: High streamline curvature and strain rate. Transitional flows. Wall heat and mass transfer. But still does not predict the spreading of a round jet correctly.

34 Improvement: realizable k-ε Shares the same turbulent kinetic energy equation as the standard k-ε model. Improved equation for ε. Variable Cµ instead of constant. Improved performance for flows involving: Planar and round jets (predicts round jet spreading correctly). Boundary layers under strong adverse pressure gradients or separation. Rotation, recirculation. Strong streamline curvature. The realizable k-ε model is a relatively recent development and differs from the standard k-epsilon model in two important ways: The realizable k-ε model contains a new formulation for the turbulent viscosity. A new transport equation for the dissipation rate, ε, has been derived from an exact equation for the transport of the mean-square vorticity fluctuation. The term "realizable'' means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows. Neither the standard k-ε model nor the RNG k-ε model is realizable.

35 k-ω model This is another two equation model. In this model ω is an inverse time scale that is associated with the turbulence. This model solves two additional PDEs: A modified version of the k equation used in the k-ε model. A transport equation for ω. The turbulent viscosity is then calculated as follows: k µt = ρ ω Its numerical behavior is similar to that of the k-ε models. It suffers from some of the same drawbacks, such as the assumption that µt is isotropic.

36 Algebraic stress model The same k and ε equations are solved as with the standard k-ε model. However, the Boussinesq assumption is not used. The full Reynolds stress equations are first derived, and then some simplifying assumptions are made that allow the derivation of algebraic equations for the Reynolds stresses. Thus fewer PDEs have to be solved than with the full RSM and it is much easier to implement. The algebraic equations themselves are not very stable, however, and computer time is significantly more than with the standard k-ε model. This model was used in the 1980s and early 1990s. Research continues but this model is rarely used in industry anymore now that most commercial CFD codes have full RSM implementations available. 38

37 Reynolds stress model RSM closes the Reynolds-Averaged Navier-Stokes equations by solving additional transport equations for the six independent Reynolds stresses. Transport equations derived by Reynolds averaging the product of the momentum equations with a fluctuating property. Closure also requires one equation for turbulent dissipation. Isotropic eddy viscosity assumption is avoided. Resulting equations contain terms that need to be modeled. RSM is good for accurately predicting complex flows. Accounts for streamline curvature, swirl, rotation and high strain rates. Cyclone flows, swirling combustor flows. Rotating flow passages, secondary flows. Flows involving separation.

38 Setting boundary conditions Characterize turbulence at inlets and outlets (potential backflow). k-ε models require k and ε. Reynolds stress model requires Rij and ε. Other options: Turbulence intensity and length scale. Length scale is related to size of large eddies that contain most of energy. For boundary layer flows, 0.4 times boundary layer thickness: l 0.4δ99. For flows downstream of grids /perforated plates: l opening size. Turbulence intensity and hydraulic diameter. Ideally suited for duct and pipe flows. Turbulence intensity and turbulent viscosity ratio. For external flows: 1 < µ t /µ < 10

39 Comparison of RANS turbulence models

40 Recommendation Start calculations by performing 100 iterations or so with standard k-ε model and first order upwind differencing. For very simple flows (no swirl or separation) converge with second order upwind and k-ε model. If the flow involves jets, separation, or moderate swirl, converge solution with the realizable k-ε model and second order differencing. If the flow is dominated by swirl (e.g. a cyclone or unbaffled stirred vessel) converge solution deeply using RSM and a second order differencing scheme. If the solution will not converge, use first order differencing instead. Ignore the existence of mixing length models and the algebraic stress model. Only use the other models if you know from other sources that somehow these are especially suitable for your particular problem (e.g. Spalart- Allmaras for certain external flows, k-ε RNG for certain transitional flows, or k-ω).

41 Tri-diagonal matrix algorithm (TDMA)

42 Consider a system of equations that has a tri-diagonal form In the above set of equations 1 and n+1 are known boundary values. The general form of any single equation is

43

44

45 End

### Numerical Methods in Aerodynamics. Turbulence Modeling. Lecture 5: Turbulence modeling

Turbulence Modeling Niels N. Sørensen Professor MSO, Ph.D. Department of Civil Engineering, Alborg University & Wind Energy Department, Risø National Laboratory Technical University of Denmark 1 Outline

### Introduction to ANSYS FLUENT

Lecture 6 Turbulence 14. 0 Release Introduction to ANSYS FLUENT 1 2011 ANSYS, Inc. January 19, 2012 Lecture Theme: Introduction The majority of engineering flows are turbulent. Successfully simulating

### Turbulence Modeling I!

Outline! Turbulence Modeling I! Grétar Tryggvason! Spring 2010! Why turbulence modeling! Reynolds Averaged Numerical Simulations! Zero and One equation models! Two equations models! Model predictions!

### Numerical Heat and Mass Transfer

Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 19 Turbulent Flows Fausto Arpino f.arpino@unicas.it Introduction All the flows encountered in the engineering practice become unstable

### 1. Introduction, tensors, kinematics

1. Introduction, tensors, kinematics Content: Introduction to fluids, Cartesian tensors, vector algebra using tensor notation, operators in tensor form, Eulerian and Lagrangian description of scalar and

### There are no simple turbulent flows

Turbulence 1 There are no simple turbulent flows Turbulent boundary layer: Instantaneous velocity field (snapshot) Ref: Prof. M. Gad-el-Hak, University of Notre Dame Prediction of turbulent flows standard

### Principles of Convection

Principles of Convection Point Conduction & convection are similar both require the presence of a material medium. But convection requires the presence of fluid motion. Heat transfer through the: Solid

### Modeling of turbulence in stirred vessels using large eddy simulation

Modeling of turbulence in stirred vessels using large eddy simulation André Bakker (presenter), Kumar Dhanasekharan, Ahmad Haidari, and Sung-Eun Kim Fluent Inc. Presented at CHISA 2002 August 25-29, Prague,

### Turbulence: Basic Physics and Engineering Modeling

DEPARTMENT OF ENERGETICS Turbulence: Basic Physics and Engineering Modeling Numerical Heat Transfer Pietro Asinari, PhD Spring 2007, TOP UIC Program: The Master of Science Degree of the University of Illinois

### Boundary-Layer Theory

Hermann Schlichting Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22

### Publication 97/2. An Introduction to Turbulence Models. Lars Davidson, lada

ublication 97/ An ntroduction to Turbulence Models Lars Davidson http://www.tfd.chalmers.se/ lada Department of Thermo and Fluid Dynamics CHALMERS UNVERSTY OF TECHNOLOGY Göteborg Sweden November 3 Nomenclature

### AER1310: TURBULENCE MODELLING 1. Introduction to Turbulent Flows C. P. T. Groth c Oxford Dictionary: disturbance, commotion, varying irregularly

1. Introduction to Turbulent Flows Coverage of this section: Definition of Turbulence Features of Turbulent Flows Numerical Modelling Challenges History of Turbulence Modelling 1 1.1 Definition of Turbulence

### Heat Transfer from An Impingement Jet onto A Heated Half-Prolate Spheroid Attached to A Heated Flat Plate

1 nd International Conference on Environment and Industrial Innovation IPCBEE vol.35 (1) (1) IACSIT Press, Singapore Heat Transfer from An Impingement Jet onto A Heated Half-Prolate Spheroid Attached to

### Turbulence - Theory and Modelling GROUP-STUDIES:

Lund Institute of Technology Department of Energy Sciences Division of Fluid Mechanics Robert Szasz, tel 046-0480 Johan Revstedt, tel 046-43 0 Turbulence - Theory and Modelling GROUP-STUDIES: Turbulence

Table of Contents Foreword.... xiii Preface... xv Chapter 1. Fundamental Equations, Dimensionless Numbers... 1 1.1. Fundamental equations... 1 1.1.1. Local equations... 1 1.1.2. Integral conservation equations...

### A Computational Investigation of a Turbulent Flow Over a Backward Facing Step with OpenFOAM

206 9th International Conference on Developments in esystems Engineering A Computational Investigation of a Turbulent Flow Over a Backward Facing Step with OpenFOAM Hayder Al-Jelawy, Stefan Kaczmarczyk

### CFD Analysis for Thermal Behavior of Turbulent Channel Flow of Different Geometry of Bottom Plate

International Journal Of Engineering Research And Development e-issn: 2278-067X, p-issn: 2278-800X, www.ijerd.com Volume 13, Issue 9 (September 2017), PP.12-19 CFD Analysis for Thermal Behavior of Turbulent

### Simulations for Enhancing Aerodynamic Designs

Simulations for Enhancing Aerodynamic Designs 2. Governing Equations and Turbulence Models by Dr. KANNAN B T, M.E (Aero), M.B.A (Airline & Airport), PhD (Aerospace Engg), Grad.Ae.S.I, M.I.E, M.I.A.Eng,

### CFD in COMSOL Multiphysics

CFD in COMSOL Multiphysics Mats Nigam Copyright 2016 COMSOL. Any of the images, text, and equations here may be copied and modified for your own internal use. All trademarks are the property of their respective

### Introduction to Turbulence AEEM Why study turbulent flows?

Introduction to Turbulence AEEM 7063-003 Dr. Peter J. Disimile UC-FEST Department of Aerospace Engineering Peter.disimile@uc.edu Intro to Turbulence: C1A Why 1 Most flows encountered in engineering and

### Manhar Dhanak Florida Atlantic University Graduate Student: Zaqie Reza

REPRESENTING PRESENCE OF SUBSURFACE CURRENT TURBINES IN OCEAN MODELS Manhar Dhanak Florida Atlantic University Graduate Student: Zaqie Reza 1 Momentum Equations 2 Effect of inclusion of Coriolis force

### Comparison of Turbulence Models in the Flow over a Backward-Facing Step Priscila Pires Araujo 1, André Luiz Tenório Rezende 2

Comparison of Turbulence Models in the Flow over a Backward-Facing Step Priscila Pires Araujo 1, André Luiz Tenório Rezende 2 Department of Mechanical and Materials Engineering, Military Engineering Institute,

### EVALUATION OF FOUR TURBULENCE MODELS IN THE INTERACTION OF MULTI BURNERS SWIRLING FLOWS

EVALUATION OF FOUR TURBULENCE MODELS IN THE INTERACTION OF MULTI BURNERS SWIRLING FLOWS A Aroussi, S Kucukgokoglan, S.J.Pickering, M.Menacer School of Mechanical, Materials, Manufacturing Engineering and

### Boundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity

Boundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity Tobias Knopp D 23. November 28 Reynolds averaged Navier-Stokes equations Consider the RANS equations with

### International Journal of Scientific & Engineering Research, Volume 6, Issue 5, May ISSN

International Journal of Scientific & Engineering Research, Volume 6, Issue 5, May-2015 28 CFD BASED HEAT TRANSFER ANALYSIS OF SOLAR AIR HEATER DUCT PROVIDED WITH ARTIFICIAL ROUGHNESS Vivek Rao, Dr. Ajay

### Detailed Outline, M E 521: Foundations of Fluid Mechanics I

Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Second-order tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic

### The effect of geometric parameters on the head loss factor in headers

Fluid Structure Interaction V 355 The effect of geometric parameters on the head loss factor in headers A. Mansourpour & S. Shayamehr Mechanical Engineering Department, Azad University of Karaj, Iran Abstract

### Computational Fluid Dynamics 2

Seite 1 Introduction Computational Fluid Dynamics 11.07.2016 Computational Fluid Dynamics 2 Turbulence effects and Particle transport Martin Pietsch Computational Biomechanics Summer Term 2016 Seite 2

### Simulating Drag Crisis for a Sphere Using Skin Friction Boundary Conditions

Simulating Drag Crisis for a Sphere Using Skin Friction Boundary Conditions Johan Hoffman May 14, 2006 Abstract In this paper we use a General Galerkin (G2) method to simulate drag crisis for a sphere,

### BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW

Proceedings of,, BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW Yunho Jang Department of Mechanical and Industrial Engineering University of Massachusetts Amherst, MA 01002 Email:

### Masters in Mechanical Engineering. Problems of incompressible viscous flow. 2µ dx y(y h)+ U h y 0 < y < h,

Masters in Mechanical Engineering Problems of incompressible viscous flow 1. Consider the laminar Couette flow between two infinite flat plates (lower plate (y = 0) with no velocity and top plate (y =

### Comparison of two equations closure turbulence models for the prediction of heat and mass transfer in a mechanically ventilated enclosure

Proceedings of 4 th ICCHMT May 17-0, 005, Paris-Cachan, FRANCE 381 Comparison of two equations closure turbulence models for the prediction of heat and mass transfer in a mechanically ventilated enclosure

### Turbulent boundary layer

Turbulent boundary layer 0. Are they so different from laminar flows? 1. Three main effects of a solid wall 2. Statistical description: equations & results 3. Mean velocity field: classical asymptotic

### The mean shear stress has both viscous and turbulent parts. In simple shear (i.e. U / y the only non-zero mean gradient):

8. TURBULENCE MODELLING 1 SPRING 2019 8.1 Eddy-viscosity models 8.2 Advanced turbulence models 8.3 Wall boundary conditions Summary References Appendix: Derivation of the turbulent kinetic energy equation

### FLUID MECHANICS. Atmosphere, Ocean. Aerodynamics. Energy conversion. Transport of heat/other. Numerous industrial processes

SG2214 Anders Dahlkild Luca Brandt FLUID MECHANICS : SG2214 Course requirements (7.5 cr.) INL 1 (3 cr.) 3 sets of home work problems (for 10 p. on written exam) 1 laboration TEN1 (4.5 cr.) 1 written exam

### Turbulence Modeling. Cuong Nguyen November 05, The incompressible Navier-Stokes equations in conservation form are u i x i

Turbulence Modeling Cuong Nguyen November 05, 2005 1 Incompressible Case 1.1 Reynolds-averaged Navier-Stokes equations The incompressible Navier-Stokes equations in conservation form are u i x i = 0 (1)

### Fluid Dynamics Exercises and questions for the course

Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r

### DEVELOPMENT OF CFD MODEL FOR A SWIRL STABILIZED SPRAY COMBUSTOR

DRAFT Proceedings of ASME IMECE: International Mechanical Engineering Conference & Exposition Chicago, Illinois Nov. 5-10, 2006 IMECE2006-14867 DEVELOPMENT OF CFD MODEL FOR A SWIRL STABILIZED SPRAY COMBUSTOR

### CHAPTER 7 NUMERICAL MODELLING OF A SPIRAL HEAT EXCHANGER USING CFD TECHNIQUE

CHAPTER 7 NUMERICAL MODELLING OF A SPIRAL HEAT EXCHANGER USING CFD TECHNIQUE In this chapter, the governing equations for the proposed numerical model with discretisation methods are presented. Spiral

### External Flow and Boundary Layer Concepts

1 2 Lecture (8) on Fayoum University External Flow and Boundary Layer Concepts By Dr. Emad M. Saad Mechanical Engineering Dept. Faculty of Engineering Fayoum University Faculty of Engineering Mechanical

### Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015

Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015 I. Introduction (Chapters 1 and 2) A. What is Fluid Mechanics? 1. What is a fluid? 2. What is mechanics? B. Classification of Fluid Flows 1. Viscous

### SG Turbulence models for CFD

SG2218 2012 Turbulence models for CFD Stefan Wallin Linné FLOW Centre Dept of Mechanics, KTH Dept. of Aeronautics and Systems Integration, FOI There are no simple turbulent flows Turbulent boundary layer:

### Turbulence is a ubiquitous phenomenon in environmental fluid mechanics that dramatically affects flow structure and mixing.

Turbulence is a ubiquitous phenomenon in environmental fluid mechanics that dramatically affects flow structure and mixing. Thus, it is very important to form both a conceptual understanding and a quantitative

### 2.3 The Turbulent Flat Plate Boundary Layer

Canonical Turbulent Flows 19 2.3 The Turbulent Flat Plate Boundary Layer The turbulent flat plate boundary layer (BL) is a particular case of the general class of flows known as boundary layer flows. The

### Explicit algebraic Reynolds stress models for internal flows

5. Double Circular Arc (DCA) cascade blade flow, problem statement The second test case deals with a DCA compressor cascade, which is considered a severe challenge for the CFD codes, due to the presence

### 7. Basics of Turbulent Flow Figure 1.

1 7. Basics of Turbulent Flow Whether a flow is laminar or turbulent depends of the relative importance of fluid friction (viscosity) and flow inertia. The ratio of inertial to viscous forces is the Reynolds

### On the transient modelling of impinging jets heat transfer. A practical approach

Turbulence, Heat and Mass Transfer 7 2012 Begell House, Inc. On the transient modelling of impinging jets heat transfer. A practical approach M. Bovo 1,2 and L. Davidson 1 1 Dept. of Applied Mechanics,

### Numerical Investigation of Thermal Performance in Cross Flow Around Square Array of Circular Cylinders

Numerical Investigation of Thermal Performance in Cross Flow Around Square Array of Circular Cylinders A. Jugal M. Panchal, B. A M Lakdawala 2 A. M. Tech student, Mechanical Engineering Department, Institute

### Chapter 6 An introduction of turbulent boundary layer

Chapter 6 An introduction of turbulent boundary layer T-S Leu May. 23, 2018 Chapter 6: An introduction of turbulent boundary layer Reading assignments: 1. White, F. M., Viscous fluid flow. McGraw-Hill,

### Before we consider two canonical turbulent flows we need a general description of turbulence.

Chapter 2 Canonical Turbulent Flows Before we consider two canonical turbulent flows we need a general description of turbulence. 2.1 A Brief Introduction to Turbulence One way of looking at turbulent

### Eddy viscosity. AdOc 4060/5060 Spring 2013 Chris Jenkins. Turbulence (video 1hr):

AdOc 4060/5060 Spring 2013 Chris Jenkins Eddy viscosity Turbulence (video 1hr): http://cosee.umaine.edu/programs/webinars/turbulence/?cfid=8452711&cftoken=36780601 Part B Surface wind stress Wind stress

### Zonal hybrid RANS-LES modeling using a Low-Reynolds-Number k ω approach

Zonal hybrid RANS-LES modeling using a Low-Reynolds-Number k ω approach S. Arvidson 1,2, L. Davidson 1, S.-H. Peng 1,3 1 Chalmers University of Technology 2 SAAB AB, Aeronautics 3 FOI, Swedish Defence

### CFD SIMULATION OF A SINGLE PHASE FLOW IN A PIPE SEPARATOR USING REYNOLDS STRESS METHOD

CFD SIMULATION OF A SINGLE PHASE FLOW IN A PIPE SEPARATOR USING REYNOLDS STRESS METHOD Eyitayo A. Afolabi 1 and J. G. M. Lee 2 1 Department of Chemical Engineering, Federal University of Technology, Minna,

### Numerical investigation of swirl flow inside a supersonic nozzle

Advances in Fluid Mechanics IX 131 Numerical investigation of swirl flow inside a supersonic nozzle E. Eslamian, H. Shirvani & A. Shirvani Faculty of Science and Technology, Anglia Ruskin University, UK

### Flow Structure Investigations in a "Tornado" Combustor

Flow Structure Investigations in a "Tornado" Combustor Igor Matveev Applied Plasma Technologies, Falls Church, Virginia, 46 Serhiy Serbin National University of Shipbuilding, Mikolayiv, Ukraine, 545 Thomas

### A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries

Center for Turbulence Research Annual Research Briefs 2006 41 A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries By D. You AND P. Moin 1. Motivation

### INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 5, ISSUE 09, SEPTEMBER 2016 ISSN

Numerical Analysis Of Heat Transfer And Fluid Flow Characteristics In Different V-Shaped Roughness Elements On The Absorber Plate Of Solar Air Heater Duct Jitesh Rana, Anshuman Silori, Rohan Ramola Abstract:

### DAY 19: Boundary Layer

DAY 19: Boundary Layer flat plate : let us neglect the shape of the leading edge for now flat plate boundary layer: in blue we highlight the region of the flow where velocity is influenced by the presence

### Turbulence Laboratory

Objective: CE 319F Elementary Mechanics of Fluids Department of Civil, Architectural and Environmental Engineering The University of Texas at Austin Turbulence Laboratory The objective of this laboratory

### Fundamentals of Fluid Dynamics: Elementary Viscous Flow

Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research

### Predictionof discharge coefficient of Venturimeter at low Reynolds numbers by analytical and CFD Method

International Journal of Engineering and Technical Research (IJETR) ISSN: 2321-0869, Volume-3, Issue-5, May 2015 Predictionof discharge coefficient of Venturimeter at low Reynolds numbers by analytical

### Basic Fluid Mechanics

Basic Fluid Mechanics Chapter 6A: Internal Incompressible Viscous Flow 4/16/2018 C6A: Internal Incompressible Viscous Flow 1 6.1 Introduction For the present chapter we will limit our study to incompressible

### Colloquium FLUID DYNAMICS 2012 Institute of Thermomechanics AS CR, v.v.i., Prague, October 24-26, 2012 p.

Colloquium FLUID DYNAMICS 212 Institute of Thermomechanics AS CR, v.v.i., Prague, October 24-26, 212 p. ON A COMPARISON OF NUMERICAL SIMULATIONS OF ATMOSPHERIC FLOW OVER COMPLEX TERRAIN T. Bodnár, L. Beneš

### HEAT TRANSFER IN A RECIRCULATION ZONE AT STEADY-STATE AND OSCILLATING CONDITIONS - THE BACK FACING STEP TEST CASE

HEAT TRANSFER IN A RECIRCULATION ZONE AT STEADY-STATE AND OSCILLATING CONDITIONS - THE BACK FACING STEP TEST CASE A.K. Pozarlik 1, D. Panara, J.B.W. Kok 1, T.H. van der Meer 1 1 Laboratory of Thermal Engineering,

### 7. TURBULENCE SPRING 2019

7. TRBLENCE SPRING 2019 7.1 What is turbulence? 7.2 Momentum transfer in laminar and turbulent flow 7.3 Turbulence notation 7.4 Effect of turbulence on the mean flow 7.5 Turbulence generation and transport

### COMPUTATIONAL SIMULATION OF THE FLOW PAST AN AIRFOIL FOR AN UNMANNED AERIAL VEHICLE

COMPUTATIONAL SIMULATION OF THE FLOW PAST AN AIRFOIL FOR AN UNMANNED AERIAL VEHICLE L. Velázquez-Araque 1 and J. Nožička 2 1 Division of Thermal fluids, Department of Mechanical Engineering, National University

### TURBULENT VORTEX SHEDDING FROM TRIANGLE CYLINDER USING THE TURBULENT BODY FORCE POTENTIAL MODEL

Proceedgs of ASME FEDSM ASME 2 Fluids Engeerg Division Summer Meetg June 11-15, 2 Boston, Massachusetts FEDSM2-11172 TURBULENT VORTEX SHEDDING FROM TRIANGLE CYLINDER USING THE TURBULENT BODY FORCE POTENTIAL

### A combined application of the integral wall model and the rough wall rescaling-recycling method

AIAA 25-299 A combined application of the integral wall model and the rough wall rescaling-recycling method X.I.A. Yang J. Sadique R. Mittal C. Meneveau Johns Hopkins University, Baltimore, MD, 228, USA

### Self-Excited Vibration in Hydraulic Ball Check Valve

Self-Excited Vibration in Hydraulic Ball Check Valve L. Grinis, V. Haslavsky, U. Tzadka Abstract This paper describes an experimental, theoretical model and numerical study of concentrated vortex flow

### Periodic planes v i+1 Top wall u i. Inlet. U m y. Jet hole. Figure 2. Schematic of computational domain.

Flow Characterization of Inclined Jet in Cross Flow for Thin Film Cooling via Large Eddy Simulation Naqavi, I.Z. 1, Savory, E. 2 and Martinuzzi, R. J. 3 1,2 The Univ. of Western Ontario, Dept. of Mech.

### CFD calculation of convective heat transfer coefficients and validation Laminar and Turbulent flow

CFD calculation of convective heat transfer coefficients and validation Laminar and Turbulent flow Subtask 3: Boundary Conditions by Adam Neale M.A.Sc. Student Concordia University Dominique Derome Concordia

### UNIT II CONVECTION HEAT TRANSFER

UNIT II CONVECTION HEAT TRANSFER Convection is the mode of heat transfer between a surface and a fluid moving over it. The energy transfer in convection is predominately due to the bulk motion of the fluid

### Validation 3. Laminar Flow Around a Circular Cylinder

Validation 3. Laminar Flow Around a Circular Cylinder 3.1 Introduction Steady and unsteady laminar flow behind a circular cylinder, representing flow around bluff bodies, has been subjected to numerous

### Turbulence Instability

Turbulence Instability 1) All flows become unstable above a certain Reynolds number. 2) At low Reynolds numbers flows are laminar. 3) For high Reynolds numbers flows are turbulent. 4) The transition occurs

### TURBULENCE MODEL AND VALIDATION OF AIR FLOW IN WIND TUNNEL

International Journal of Technology (2016) 8: 1362-1372 ISSN 2086-9614 IJTech 2016 TURBULENCE MODEL AND VALIDATION OF AIR FLOW IN WIND TUNNEL Gun Gun Ramdlan G. 1,2*, Ahmad Indra Siswantara 1, Budiarso

### Engineering. Spring Department of Fluid Mechanics, Budapest University of Technology and Economics. Large-Eddy Simulation in Mechanical

Outline Geurts Book Department of Fluid Mechanics, Budapest University of Technology and Economics Spring 2013 Outline Outline Geurts Book 1 Geurts Book Origin This lecture is strongly based on the book:

### Introduction to CFD modelling of source terms and local-scale atmospheric dispersion (Part 1 of 2)

1 Introduction to CFD modelling of source terms and local-scale atmospheric dispersion (Part 1 of 2) Atmospheric Dispersion Modelling Liaison Committee (ADMLC) meeting 15 February 2018 Simon Gant, Fluid

### Chapter 8 Flow in Conduits

57:00 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 013 1 Chapter 8 Flow in Conduits Entrance and developed flows Le = f(d, V,, ) i theorem Le/D = f(re) Laminar flow:

### CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION

CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,

### RECONSTRUCTION OF TURBULENT FLUCTUATIONS FOR HYBRID RANS/LES SIMULATIONS USING A SYNTHETIC-EDDY METHOD

RECONSTRUCTION OF TURBULENT FLUCTUATIONS FOR HYBRID RANS/LES SIMULATIONS USING A SYNTHETIC-EDDY METHOD N. Jarrin 1, A. Revell 1, R. Prosser 1 and D. Laurence 1,2 1 School of MACE, the University of Manchester,

### On the feasibility of merging LES with RANS for the near-wall region of attached turbulent flows

Center for Turbulence Research Annual Research Briefs 1998 267 On the feasibility of merging LES with RANS for the near-wall region of attached turbulent flows By Jeffrey S. Baggett 1. Motivation and objectives

### 2. FLUID-FLOW EQUATIONS SPRING 2019

2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid

### Calculations on a heated cylinder case

Calculations on a heated cylinder case J. C. Uribe and D. Laurence 1 Introduction In order to evaluate the wall functions in version 1.3 of Code Saturne, a heated cylinder case has been chosen. The case

### Modeling Complex Flows! Direct Numerical Simulations! Computational Fluid Dynamics!

http://www.nd.edu/~gtryggva/cfd-course/! Modeling Complex Flows! Grétar Tryggvason! Spring 2011! Direct Numerical Simulations! In direct numerical simulations the full unsteady Navier-Stokes equations

### Chapter 8: Flow in Pipes

8-1 Introduction 8-2 Laminar and Turbulent Flows 8-3 The Entrance Region 8-4 Laminar Flow in Pipes 8-5 Turbulent Flow in Pipes 8-6 Fully Developed Pipe Flow 8-7 Minor Losses 8-8 Piping Networks and Pump

### WALL ROUGHNESS EFFECTS ON SHOCK BOUNDARY LAYER INTERACTION FLOWS

ISSN (Online) : 2319-8753 ISSN (Print) : 2347-6710 International Journal of Innovative Research in Science, Engineering and Technology An ISO 3297: 2007 Certified Organization, Volume 2, Special Issue

### We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists. International authors and editors

We are IntechOpen, the world s leading publisher of Open Access books Built by scientists, for scientists 3,800 116,000 120M Open access books available International authors and editors Downloads Our

### Visualization of flow pattern over or around immersed objects in open channel flow.

EXPERIMENT SEVEN: FLOW VISUALIZATION AND ANALYSIS I OBJECTIVE OF THE EXPERIMENT: Visualization of flow pattern over or around immersed objects in open channel flow. II THEORY AND EQUATION: Open channel:

### Introduction to Turbulence and Turbulence Modeling

Introduction to Turbulence and Turbulence Modeling Part I Venkat Raman The University of Texas at Austin Lecture notes based on the book Turbulent Flows by S. B. Pope Turbulent Flows Turbulent flows Commonly

### The current issue and full text archive of this journal is available at

The current issue and full text archive of this journal is available at www.emeraldinsight.com/0961-5539.htm HFF 16,6 660 Received February 2005 Revised December 2005 Accepted December 2005 3D unsteady

### HEAT TRANSFER BY CONVECTION. Dr. Şaziye Balku 1

HEAT TRANSFER BY CONVECTION Dr. Şaziye Balku 1 CONDUCTION Mechanism of heat transfer through a solid or fluid in the absence any fluid motion. CONVECTION Mechanism of heat transfer through a fluid in the

### ρ t + (ρu j ) = 0 (2.1) x j +U j = 0 (2.3) ρ +ρ U j ρ

Chapter 2 Mathematical Models The following sections present the equations which are used in the numerical simulations documented in this thesis. For clarity, equations have been presented in Cartesian

### OpenFOAM selected solver

OpenFOAM selected solver Roberto Pieri - SCS Italy 16-18 June 2014 Introduction to Navier-Stokes equations and RANS Turbulence modelling Numeric discretization Navier-Stokes equations Convective term {}}{

### Module 3: Velocity Measurement Lecture 15: Processing velocity vectors. The Lecture Contains: Data Analysis from Velocity Vectors

The Lecture Contains: Data Analysis from Velocity Vectors Velocity Differentials Vorticity and Circulation RMS Velocity Drag Coefficient Streamlines Turbulent Kinetic Energy Budget file:///g /optical_measurement/lecture15/15_1.htm[5/7/2012

### TURBULENT FLOW ACROSS A ROTATING CYLINDER WITH SURFACE ROUGHNESS

HEFAT2014 10 th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics 14 16 July 2014 Orlando, Florida TURBULENT FLOW ACROSS A ROTATING CYLINDER WITH SURFACE ROUGHNESS Everts, M.,

### ABSTRACT OF ONE-EQUATION NEAR-WALL TURBULENCE MODELS. Ricardo Heinrich Diaz, Doctor of Philosophy, 2003

ABSTRACT Title of dissertation: CRITICAL EVALUATION AND DEVELOPMENT OF ONE-EQUATION NEAR-WALL TURBULENCE MODELS Ricardo Heinrich Diaz, Doctor of Philosophy, 2003 Dissertation directed by: Professor Jewel

### CFD STUDY OF MASS TRANSFER IN SPACER FILLED MEMBRANE MODULE

GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 33-41 CFD STUDY OF MASS TRANSFER IN SPACER FILLED MEMBRANE MODULE Sharmina Hussain Department of Mathematics and Natural Science BRAC University,

### 4.2 Concepts of the Boundary Layer Theory

Advanced Heat by Amir Faghri, Yuwen Zhang, and John R. Howell 4.2 Concepts of the Boundary Layer Theory It is difficult to solve the complete viscous flow fluid around a body unless the geometry is very