1 intro turbulent flows

AER1310: TURBULENCE MODELLING 1. Introduction to Turbulent Flows C. P. T. Groth c©2015

1. Introduction to Turbulent Flows

Coverage of this section:

I Definition of Turbulence

I Features of Turbulent Flows

I Numerical Modelling Challenges

I History of Turbulence Modelling

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1.1 Definition of Turbulence

Oxford Dictionary: disturbance, commotion, varying irregularly

Webster’s Dictionary: agitation, commotion, erratic velocity

Taylor and Von Karman (1937): “Turbulence is an irregular motionwhich in general makes its appearance in fluids, gaseous or liquid,when they flow past solid surfaces or even when neighbouringstreams of the same fluid flow past or over one another.”

There are problems with the Taylor-Von Karman definition:

I not sufficient to say that turbulence is associated withirregular motion; and

I there are non-turbulent flows that can be described asirregular.

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Hinze (1959): “Turbulent fluid motion is an irregular condition offlow in which the various quantities show a random variation withtime and space coordinates, so that statistically distinct averagevalues can be discerned.”

From Hinze’s definition it should be noted that:

I instantaneous flow is sensitive to initial conditions but thestatistical averages are not; and

I it is not sufficient to define turbulent motion as irregular intime alone.

Bradshaw (1974): “Turbulence has a wide range of scales”.

CPTG (2003): “Inherently three-dimensional and time dependent.”

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1.2 Features of Turbulent Flows

1.2.1 Important to a Wide Range of Fields

Virtually all flows of practical interest are turbulent:

I flow past vehicles such as automobiles, airplanes, ships, &rockets;

I flows associated with power generation & propulsion (i.e., gasturbine engines); and

I geophysical and atmospheric flow applications such as rivercurrents and motion of clouds.

In all of these applications, the flows of interest are predominantlyturbulent.

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1.2.2 Dependence on Reynolds Number

In contrast to turbulent flow, laminar flow structure appearslayered with well defined streamline structure (fluid laminae).Turbulence disrupts the layered structure.

The boundary between laminar and turbulent flow regimes iseffectively defined by the flow Reynolds number, Re, which is theratio of inertial forces to viscous surface forces and given by

Re =ρu`

µ=

u`

ν

where ρ is the fluid density, u is the flow velocity, µ is the dynamicviscosity, ν=µ/ρ is the kinematic viscosity, and ` is thecharacteristic length scale of interest. Flows tend to becometurbulent as Re becomes large.

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Turbulent Pipe Flow (Recrit≈2, 300)

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Turbulent Pipe Flow (Recrit≈2, 300)

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1.2 Features of Turbulent FlowsTurbulent Flow Past a Flat Plate (Rexcrit≈320, 000,Reδcrit≈2, 800)

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1.2 Features of Turbulent FlowsTurbulent Flow Past a Flat Plate (Rexcrit≈320, 000,Reδcrit≈2, 800)

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1.2.3 Turbulent Vs. Laminar Flows

The behaviour of laminar and turbulent flows are very distinct.The important effects of turbulent motion include:

I Enhanced Diffusivity: turbulent diffusion greatly enhances thetransfer (transport) of mass, momentum, and energy.Apparent fluid stresses are several orders of magnitude largerthan in the corresponding laminar flow.

I Increased Skin Drag: turbulent boundary layer velocity profilesare generally thicker and more “full” and this increases theviscous drag as surfaces.

I Less Susceptible to Flow Separation: turbulent boundarylayers are less likely to separate and can support strongeradverse pressure gradients while laminar boundary layersgenerally cannot support even mild adverse pressure gradients.

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1.2.3 Turbulent Vs. Laminar Flows

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Turbulent Boundary Layer Profiles

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1.2.4 Instability and Nonlinearity

I Transition from laminar to turbulent flow is due to nonlinearinstabilities of the Navier-Stokes equations.

I Instabilities result mainly from the interaction between thenonlinear inertial and viscous terms.

I Inviscid instabilities (i.e., Kelvin-Helmholtz instabilities) alsoplay a role.

I Linear stability analysis of boundary layer equations(Orr-Summerfield equations) predicts growth modes andinstability of laminar flows but cannot accurately predict thetransition from laminar to turbulent flow.

I Understanding and predicting transition prediction is beyondthe scope and not the focus of this course. We will generallyassume that the flow is fully turbulent.

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Transition to Turbulence for a Flat Plate

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1.2.5 Turbulent Eddies

Turbulence consist of a continuous spectrum of scales ranging fromthe largest to the smallest scales. It can be thought to consist ofturbulent eddies of varying sizes, where u is the eddy velocity scale,` is the eddy length scale, and a eddy time scale, τ , can be definedas τ =`/u. The eddies overlap in space with larger ones carryingsmaller ones.

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Energy Cascade

Due to instabilities, the large eddies eventually break up, producingsuccessively smaller eddies. The kinetic energy of the larger eddiesis divided among the smaller eddies. This process is repeated downto the small scales. This leads to an energy cascade in whichenergy is passed down from the large scales to smaller scales whereeventually the kinetic energy is dissipated as heat.

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Richardson, 1922

This notion that that a turbulent flow is composed of a ”cascadeof eddies” of different sizes is an idea that was orginally introducedby Lewis Richardson in 1922. He composed the following rhymingverse 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.

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Large-Scale Eddies

In general, the large-scale eddies contain most of the turbulentkinetic energy (kinetic energy associated with turbulent motion)and are mainly responsible for the enhanced diffusivity andincreased apparent stresses.

The large scales, as represented by the integral length scale, arealso generally not statistically isotropic (i.e., having no preferentialspatial direction), since they are determined by the particulargeometrical features of the flow and its boundaries.

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Large-Scale Eddies

For free-shear flows, the size of the largest eddies, `, is of order

` ∝ δ (thickness of shear layer)

and, for wall-bounded flows, the largest scales are of order

` ∝ y (distance from the wall)

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Small-Scale Eddies

The smallest scale eddies are at the Kolmogorov scales, η. This isthe smallest scale at which the turbulence can exist. The energy inthe turbulent motion (i.e., the turbulent kinetic energy) isdissipated as heat by molecular viscosity at the Kolmogorov scales.Most of the vorticity of a turbulent flow resides in the smallesteddies.

Turbulence therefore consists of a continuous range of scales fromthe largest energy-carrying scales, `, to the smallest Kolmogorovscales, η, with a large separation of these scales, i.e.,

`

η� 1

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Taylor Micro-Scale

The Taylor micro-scale, `T, is an intermediate scale between thelargest and the smallest turbulence scales. It typically lies withinthe so-called inertial subrange but well above the Kolmogorovscale. The Taylor micro-scale can be approximated by

`Tη≈ 7

(`

η

)(1/3)

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1.2.6 Kolmogorov Scales

Estimates of the Kolmogorov scale can be found by applyingKolmogorov’s universal equilibrium theory (1941) and equating therate of energy transfer from the large scales to the rate ofdissipation of turbulent energy to heat by molecular viscosity, ν, atthe small scales, i.e.,

dk

dt= −ε

where k is the turbulent kinetic energy and ε is the dissipationrate. Using dimensional analysis, it then follows that

η ≡(ν3

ε

)1/4

(Kolmogorov length scale)

τ ≡(νε

)1/2υ ≡ (νε)1/4 (Kolmogorov time & velocity scales)

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1.2.7 Continuum Phenomenon“Even the smallest scales occurring in a turbulent flow areordinarily far larger than any molecular length scale” (Tennekes &Lumley, 1983).Consider the Knudsen number, Kn=λ/η, for the small scaleswhere λ is the mean free path for the gas (average distancetravelled by gaseous particles between collisions). Assuming thatν≈(1/2)cλ and using c =

√8kT/πm then

Kn ≈ 2ν

ηc≤ 0.01

This implies that the continuum approximation (i.e., theNavier-Stokes equations) are fully valid down to the Kolmogorovscales.

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1.2.8 Homogeneous & Isotropic Turbulence

Energy Cascade: As noted, turbulence features a cascade processwhereby, as the turbulence decays, kinetic energy is transferredfrom larger to smaller eddies until it is dissipated at the smallestscales.

Dissipative Process: Furthermore, turbulence is dissipative innature and without a continuous source of external energy for thegeneration of turbulence, the motion will decay.

The energy cascade and dissipation of energy has a strongtendency to make the turbulence more homogeneous and isotropic.

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Homogeneous Turbulence: turbulent flow that has statistically thesame structure in all parts of the flow field.

Isotropic Turbulence: turbulent flow whose statistical features haveno preference for a spatial coordinate direction.

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Kolmogorov, 1941

Kolmogorov postulated that, for very high Reynolds numbers, thethe small scale turbulent motions become statistically isotropic(i.e. having no preferential spatial direction). Through the energycascade, the geometrical and directional information of thegenerally anisotropic larger scales is lost as the scale is reduced, sothat the statistics of the small scales become more isotropic and,when the Reynolds number is sufficiently high, they eventuallyachieve a universal character, the same for all turbulent flows. Thebehaviour of these universal small scales is then uniquelydetermined by the viscosity, ν, and the rate of energy dissipation, ε.

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1.3 Numerical Modelling Challenges

1.3.1 Difficulty of Calculating Turbulent Flows

Questions: The continuum assumption applies and theNavier-Stokes equations provide a complete description ofturbulence, so why not just solve the equations directly from firstprinciples (i.e., using a Direct Numerical Simulation (DNS)technique) and be done with it? Why bother with turbulencemodels?

Answers: Performing DNS of turbulence is a very difficultchallenge for the following reasons:

I turbulent flow is inherently 3D and time dependent; and

I all physically relevant scales down to the Kolmogorov scalemust be resolved.

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1.3.1 Difficulty of Calculating Turbulent Flows

Example: Consider turbulent flow over a slender airfoil-like bodywith u=50 m/s and a body length of L=9 m. In order to resolveall of the necessary scales, it is estimated that a computationalmesh of size

N = 20, 000× 1, 200× 4, 800 = 115× 109 = 115 billion nodes

would be required. Even for this relatively low velocity and simplegeometry, the problem is currently impossible to solve using DNS.DNS is reserved for model flow problems of academic interest forunderstanding fundamentals of turbulent flows. Generally limitedto flows with simple geometries, periodic boundaries, etc...DNS cannot currently nor will it in the near future be used topredict practical engineering flows!

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1.3 Numerical Modelling Challenges

1.3.2 Turbulence Models

Turbulence Models: Provide approximate descriptions of turbulenceand “should introduce the minimum amount of complexity whilecapturing the essence of the relevant physics” (Wilcox, 2002).

Turbulence modelling is one of the key elements of computationalfluid dynamics (CFD). It enables the solution of practicalengineering flows.

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1.4 History of Turbulence Modelling

History of turbulence modelling dates back more than 100 years:

I 1877 – Boussinesq – eddy viscosity concept

I 1895 – Reynolds – Reynolds averaging

I 1904 – Prandtl – boundary layer

I 1925 – Prandtl – mixing length model

I 1930 – Von Karman – early turbulence research

I 1942 – Kolmogorov – two-equation model

I 1945 – Prandtl – k-equation and one-equation model

I 1945 – Chow – second-order Reynolds-stress closure

I 1951 – Rott – second-order Reynolds-stress closure

I 1956 – Van Driest – algebraic model

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I 1963 – Smagorinski – subgrid-scale LES model

I 1968 – Donaldson – second-order Reynolds-stress closure

I 1969 – Wolfstein – one-equation model

I 1970 – Daly & Harlow – second-order Reynolds-stress closure

I 1972 – Launder & Spalding – two-equation (k-ε) model

I 1974 – Cebeci & Smith – algebraic model

I 1975 – Launder, Reece, & Rodi – second-orderReynolds-stress closure

I 1978 – Baldwin & Lomax – algebraic model

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I 1988 – Wilcox – two-equation (k-ω) model

I 1990 – Baldwin & Barth – one-equation model

I 1991 – Germano, Piomelli, Moin, & Cabot – dynamic subgridscale model

I 1992 – Spalart & Allmaras – one-equation model

I 1994 – Menter – two-equation (SST) model

I 1990s & 2000s – LES, DES, & DNS

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1 intro turbulent flows

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