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

Size: px

Start display at page:

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

Sherilyn Janice Haynes
6 years ago
Views:

1 1. Introduction to Turbulent Flows Coverage of this section: Definition of Turbulence Features of Turbulent Flows Numerical Modelling Challenges History of Turbulence Modelling Definition of Turbulence Oxford Dictionary: disturbance, commotion, varying irregularly Webster s Dictionary: agitation, commotion, erratic velocity Taylor and Von Kármán (1937): Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighbouring streams of the same fluid flow past or over one another. There are problems with the Taylor-Von Kármán definition: not sufficient to say that turbulence is associated with irregular motion; and there are non-turbulent flows that can be described as irregular. 2

AER1310: TURBULENCE MODELLING 1. Introduction to Turbulent Flows C. P. T. Groth c 2018 1.

statistically distinct average values can be discerned.

2 AER1310: TURBULENCE MODELLING 1. Introduction to Turbulent Flows C. P. T. Groth c Definition of Turbulence Hinze (1959): Turbulent fluid motion is an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned. From Hinze s definition it should be noted that: I instantaneous flow is sensitive to initial conditions but the statistical averages are not; and I it is not sufficient to define turbulent motion as irregular in time alone. Bradshaw (1974): Turbulence has a wide range of scales. CPTG (2003): Inherently three-dimensional and time dependent. 3 AER1310: TURBULENCE MODELLING 1. Introduction to Turbulent Flows 1.1 Definition of Turbulence 4 C. P. T. Groth c 2018

3 1.2.1 Important to a Wide Range of Fields Virtually all flows of practical interest are turbulent: flow past vehicles such as automobiles, airplanes, ships, & rockets; flows associated with power generation & propulsion (i.e., gas turbine engines); and geophysical and atmospheric flow applications such as river currents and motion of clouds. In all of these applications, the flows of interest are predominantly turbulent Dependence on Reynolds Number In contrast to turbulent flow, laminar flow structure appears layered with well defined streamline structure (fluid laminae). Turbulence disrupts the layered structure. The boundary between laminar and turbulent flow regimes is effectively defined by the flow Reynolds number, Re, which is the ratio of inertial forces to viscous surface forces and given by Re = ρul µ = ul ν where ρ is the fluid density, u is the flow velocity, µ is the dynamic viscosity, ν =µ/ρ is the kinematic viscosity, and l is the characteristic length scale of interest. Flows tend to become turbulent as Re becomes large. 6

4 Turbulent Pipe Flow (Re crit 2, 300) 7 Turbulent Pipe Flow (Re crit 2, 300) 8

5 Turbulent Flow Past a Flat Plate (Re xcrit 320, 000, Re δcrit 2, 800) 9 Turbulent Flow Past a Flat Plate (Re xcrit 320, 000, Re δcrit 2, 800) 10

6 1.2.3 Turbulent Vs. Laminar Flows The behaviour of laminar and turbulent flows are very distinct. The important effects of turbulent motion include: Enhanced Diffusivity: turbulent diffusion greatly enhances the transfer (transport) of mass, momentum, and energy. Apparent fluid stresses are several orders of magnitude larger than in the corresponding laminar flow. Increased Skin Drag: turbulent boundary layer velocity profiles are generally thicker and more full and this increases the viscous drag as surfaces. Less Susceptible to Flow Separation: turbulent boundary layers are less likely to separate and can support stronger adverse pressure gradients while laminar boundary layers generally cannot support even mild adverse pressure gradients Turbulent Vs. Laminar Flows 12

Turbulent Boundary Layer Profiles 13 1.2.4 Instability and Nonlinearity Transition from laminar to turbulent flow is due to nonlinear instabilities of the Navier-Stokes equations.

7 Turbulent Boundary Layer Profiles Instability and Nonlinearity Transition from laminar to turbulent flow is due to nonlinear instabilities of the Navier-Stokes equations. Instabilities result mainly from the interaction between the nonlinear inertial and viscous terms. Inviscid instabilities (i.e., Kelvin-Helmholtz instabilities) also play a role. Linear stability analysis of boundary layer equations (Orr-Summerfield equations) predicts growth modes and instability of laminar flows but cannot accurately predict the transition from laminar to turbulent flow. Understanding and predicting transition prediction is beyond the scope and not the focus of this course. We will generally assume that the flow is fully turbulent. 14

Transition to Turbulence for a Flat Plate 15 1.2.5 Turbulent Eddies Turbulence consist of a continuous spectrum of scales ranging from the largest to the smallest scales.

8 Transition to Turbulence for a Flat Plate Turbulent Eddies Turbulence consist of a continuous spectrum of scales ranging from the largest to the smallest scales. It can be thought to consist of turbulent eddies of varying sizes, where u is the eddy velocity scale, l is the eddy length scale, and a eddy time scale, τ, can be defined as τ =l/u. The eddies overlap in space with larger ones carrying smaller ones. 16

9 1.2.5 Turbulent Eddies Energy Cascade Due to instabilities, the large eddies eventually break up, producing successively smaller eddies. The kinetic energy of the larger eddies is divided among the smaller eddies. This process is repeated down to the small scales. This leads to an energy cascade in which energy is passed down from the large scales to smaller scales where eventually the kinetic energy is dissipated as heat Turbulent Eddies Richardson, 1922 This notion that a turbulent flow is composed of a cascade of eddies of different sizes is an idea that was orginally introduced by Lewis Richardson in He composed the following rhyming verse that captures this viewpoint: Big whorls have little whorls, Which feed on their velocity; And little whorls have lesser whorls, And so on to viscosity. 18

10 1.2.5 Turbulent Eddies Large-Scale Eddies In general, the large-scale eddies contain most of the turbulent kinetic energy (kinetic energy associated with turbulent motion) and are mainly responsible for the enhanced diffusivity and increased apparent stresses. The large scales, as represented by the integral length scale, are also generally not statistically isotropic (i.e., having no preferential spatial direction), since they are determined by the particular geometrical features of the flow and its boundaries Turbulent Eddies Large-Scale Eddies For free-shear flows, the size of the largest eddies, l, is of order l δ (thickness of shear layer) and, for wall-bounded flows, the largest scales are of order l y (distance from the wall) 20

11 1.2.5 Turbulent Eddies Small-Scale Eddies The smallest scale eddies are at the Kolmogorov scales, η. This is the smallest scale at which the turbulence can exist. The energy in the turbulent motion (i.e., the turbulent kinetic energy) is dissipated as heat by molecular viscosity at the Kolmogorov scales. Most of the vorticity of a turbulent flow resides in the smallest eddies. Turbulence therefore consists of a continuous range of scales from the largest energy-carrying scales, l, to the smallest Kolmogorov scales, η, with a large separation of these scales, i.e., l η Taylor Micro Scales The Taylor micro scale, l T, is an intermediate scale between the largest and the smallest turbulence scales. It typically lies within the so-called inertial subrange, as defined by Kolmogorov s second similarity hypothesis, but well above the Kolmogorov scale. The Taylor micro-scale can be approximated by l T η 7 ( l η ) (1/3) It is argued that this is the intermediate length scale at which fluid viscosity begins to have significant affects on the dynamics of turbulent eddies in the flow. Turbulent length scales which are larger than the Taylor microscale are not strongly affected by viscosity. 22

12 1.2.7 Kolmogorov Scales Estimates of the Kolmogorov scale can be found by applying Kolmogorov s universal equilibrium theory (1941). This theory is actually based on three hypotheses: (i) Kolmogorov s hypothesis of local isotropy; (ii) Kolmogorov s first-similarity hypothesis defining the smallest scalest of turbulence; and (iii) Kolmogorov s second-similarity hypothesis defining the inertial subrange and is backed up by both dimensional arguments and experimental observations Kolmogorov Scales By equating the rate of energy transfer from the large scales to the rate of dissipation of turbulent energy to heat by molecular viscosity, ν, at the small scales, i.e., dk dt = ɛ where k is the turbulent kinetic energy and ɛ is the dissipation rate. Using dimensional analysis, it then follows that ( ) ν 3 1/4 η (Kolmogorov length scale) ɛ ( ν ) 1/2 τ υ (νɛ) 1/4 (Kolmogorov time & velocity scales) ɛ 24

13 1.2.8 Continuum Phenomenon Even the smallest scales occurring in a turbulent flow are ordinarily far larger than any molecular length scale (Tennekes & Lumley, 1983). Consider the Knudsen number, Kn=λ/η, for the small scales where λ is the mean free path for the gas (average distance travelled by gaseous particles between collisions). Assuming that ν (1/2) cλ and using c = 8kT /πm then Kn 2ν η c 0.01 This implies that the continuum approximation (i.e., the Navier-Stokes equations) are fully valid down to the Kolmogorov scales Homogeneous & Isotropic Turbulence Energy Cascade: As noted, turbulence features a cascade process whereby, as the turbulence decays, kinetic energy is transferred from larger to smaller eddies until it is dissipated at the smallest scales. Dissipative Process: Furthermore, turbulence is dissipative in nature and without a continuous source of external energy for the generation of turbulence, the motion will decay. The energy cascade and dissipation of energy has a strong tendency to make the turbulence more homogeneous and isotropic. 26

14 1.2.9 Homogeneous & Isotropic Turbulence Homogeneous Turbulence: turbulent flow that has statistically the same structure in all parts of the flow field. Isotropic Turbulence: turbulent flow whose statistical features have no preference for a spatial coordinate direction Homogeneous & Isotropic Turbulence Kolmogorov, 1941 Kolmogorov postulated that, for very high Reynolds numbers, the the small scale turbulent motions become statistically isotropic (i.e. having no preferential spatial direction). This is the basis for Kolmogorov s hypothesis of local isotropy. Through the energy cascade, the geometrical and directional information of the generally anisotropic larger scales is lost as the scale is reduced, so that the statistics of the small scales become more isotropic and, when the Reynolds number is sufficiently high, they eventually achieve a universal character, the same for all turbulent flows. The behaviour of these universal small scales is then uniquely determined by the viscosity, ν, and the rate of energy dissipation, ɛ. These ideas are the basis for Kolmogorov s first similarity hypothesis. 28

15 1.3 Numerical Modelling Challenges Difficulty of Calculating Turbulent Flows Questions: The continuum assumption applies and the Navier-Stokes equations provide a complete description of turbulence, so why not just solve the equations directly from first principles (i.e., using a Direct Numerical Simulation (DNS) technique) and be done with it? Why bother with turbulence models? Answers: Performing DNS of turbulence is a very difficult challenge for the following reasons: turbulent flow is inherently 3D and time dependent; and all physically relevant scales down to the Kolmogorov scale must be resolved Difficulty of Calculating Turbulent Flows Example: Consider turbulent flow over a slender airfoil-like body with u =50 m/s and a body length of L=9 m. In order to resolve all of the necessary scales, it is estimated that a computational mesh of size N = 20, 000 1, 200 4, 800 = = 115 billion nodes would be required. Even for this relatively low velocity and simple geometry, the problem is currently impossible to solve using DNS. DNS is reserved for model flow problems of academic interest for understanding fundamentals of turbulent flows. Generally limited to flows with simple geometries, periodic boundaries, etc... DNS cannot currently nor will it in the near future be used to predict practical engineering flows! 30

16 1.3 Numerical Modelling Challenges Turbulence Models Turbulence Models: Provide approximate descriptions of turbulence and should introduce the minimum amount of complexity while capturing the essence of the relevant physics (Wilcox, 2002). Turbulence modelling is one of the key elements of computational fluid dynamics (CFD). It enables the solution of practical engineering flows History of Turbulence Modelling History of turbulence modelling dates back more than 100 years: 1877 Boussinesq eddy viscosity concept 1895 Reynolds Reynolds averaging 1904 Prandtl boundary layer 1925 Prandtl mixing length model 1930 Von Kármán early turbulence research 1942 Kolmogorov two-equation model 1945 Prandtl k-equation and one-equation model 1945 Chow second-order Reynolds-stress closure 1951 Rott second-order Reynolds-stress closure 1956 Van Driest algebraic model 32

17 1.4 History of Turbulence Modelling 1963 Smagorinski subgrid-scale LES model 1968 Donaldson second-order Reynolds-stress closure 1969 Wolfstein one-equation model 1970 Daly & Harlow second-order Reynolds-stress closure 1972 Launder & Spalding two-equation (k-ɛ) model 1974 Cebeci & Smith algebraic model 1975 Launder, Reece, & Rodi second-order Reynolds-stress closure 1978 Baldwin & Lomax algebraic model History of Turbulence Modelling 1988 Wilcox two-equation (k-ω) model 1990 Baldwin & Barth one-equation model 1991 Germano, Piomelli, Moin, & Cabot dynamic subgrid scale model 1992 Spalart & Allmaras one-equation model 1994 Menter two-equation (SST) model 1990s & 2000s LES, DES, & DNS 34

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

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