Cambridge University Engineering Department
4A12: Turbulence and Vortex Dynamics
Part II: Turbulence
Lecture Notes
Prof. E. Mastorakos
Hopkinson Lab
E-mail: [email protected]
http://www.eng.cam.ac.uk/∼em257
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Cambridge University Engineering Department
4A12: Turbulence and Vortex Dynamics
Part II: Turbulence
Lecture Notes
Prof. E. Mastorakos Hopkinson Lab E-mail: [email protected] http://www.eng.cam.ac.uk/∼em
Chapter 1
Introduction
1.1 Aims
The main aims of this course are:
- To introduce some fundamental physical aspects of turbulence.
- To describe the phenomenology of some turbulent flows.
- To allow the student make simple estimates of the order of magnitude of certain quan- tities in turbulent flows.
1.2 Introduction
Turbulent flows are found everywhere and we all have some intuitive feeling as to what constitutes a turbulent flow. A good way to describe the various facets of turbulent motion is to consider it as a “syndrome”, with some special characteristics (Stewart, 1969, NSF Film). These include disorder, vorticity, and mixing. Tennekes and Lumley [1] in the Introduction to their book, mention irregularity, diffusivity, vorticity fluctuations, dissipation and large Reynolds numbers as a few of the key features of the nature of turbulence. Turbulence is created through instabilities of laminar flows at large Re, e.g. laminar pipe flow becomes turbulent at about Re = U d/ν = 2000 and a boundary layer at a U δ∗/ν = 600 (δ∗^ is the displacement thickness). Most turbulent flows involve velocity shear; if shear or any other production mechanism such as buoyancy are absent, then turbulence decays. We will discuss later the rate of this decay and on how to estimate the velocity fluctuations. There is no general approach to the solution of turbulent flows, but one of the most prominent methods is to analyze the equations of motion or experimental data in a statistical manner. Often, the time-averaged values of the main quantities (first-order moments) is the only thing we are interested in from an engineering point of view. However, there are many situations where we are also interested in relatively rare events, e.g. a high temperature excursion from the mean in a flame or a high-concentration parcel of fluid in a pollutant plume, where we would like to be in a position to estimate the full range of possible values. This we achieve with probability densities and by considering higher moments. Statistical
Bibliography
[1] H. Tennekes and J.L. Lumley. A first course in turbulence. MIT Press. A classic. Excellent for all parts of this course. Chapter 1 of this book must be read by all students of turbulence.
[2] P.A. Davidson. Turbulence. An introduction for scientists and engineers. Oxford Uni- versity Press, 2004. A thorough exposition. Chapters 1 and 4 are especially relevant for us.
[3] S.B. Pope. Turbulent flows. Cambridge University Press, 2000. Chapter 1, Chapter 4 (for the equations), Chapter 5 (jets, wakes, shear flow), Chapter 6 (pp. 182-190) for the energy cascade.
[4] P.A. Durbin and B.A. Pettersson Reif. Statistical theory and modelling for turbulent flows. Wiley, 2001. Chapter 1 for an introduction, Chapter 2 for the mathematical tools necessary in the study of turbulence, Chapter 3 for the equations, Chapter 4 for free flows and boundary layers.
[5] W.D. McComb. The physics of fluid turbulence. Oxford University Press, 1990. Chapter 1 covers briefly most of the material in our course. Chapter 12 (pp. 436-441) is very appropriate for our “Turbulent transport” analysis.
[6] A.A. Townsend. The structure of turbulent shear flow. Cambridge University Press, second edition, 1976. Chapter 2 for the governing equations, Chapter 6 for free flows, Chapter 7 for boundary layers. Slightly too detailed for our level.
[7] J. Hinze. Turbulence. Mc Graw Hill, second edition, 1975. Parts of Chapter 1 for the governing equations and correlations.
[8] H. Schlichting. Boundary-layer theory. Mc Graw Hill, seventh edition, 1979. Chapters 18 and 19 for a brief exposition of the basics and the governing equations. Chapter 21 for the flat plate boundary layer.
[9] G.T. Csanady. Turbulent diffusion in the environment. Reidel Publishing Company,
- For our Chapter 4.
Chapter 2
Statistical description and scales
2.1 Statistical description
2.1.1 Averages and randomness
Figure 2.1 shows a hypothetical velocity signal measured with a fast-response probe (e.g. a hot wire) in turbulent flow. This time-series could also be that of a scalar. We can associate an average value u with this signal, where the time-average is defined by:
u = lim T →∞
T
∫ T
0
u(t)dt (2.1)
In practice, the upper limit T would be a finite time, which should be long enough for the integral to converge. This definition is valid if the time-average is independent of time. Otherwise, u should be thought of as an ensemble average compiled from many different realizations of the flow. So, if we have N realizations,
u =
N
∑^ N
i=
ui (2.2)
We will discuss later explicitly how large T or N should be for the average to be meaningful. The fact that turbulence has randomness implies that, other than by solving the full Navier-Stokes equations, there is no way of getting u(t + ∆t) from u(t), where ∆t is a finite time increment. Pope [3] defines a random event as one for which we cannot say with certainty that it will occur or that it will not occur, for example the event being that u is, say, between 10 and 11 m/s. A random signal is usually interpreted in terms of its mean, but a measure of the fluctuations (i.e. the deviations of the signal in Fig. 2.1 about u) must also be provided.
2.1.2 Probability density function
As with all random variables (or processes), we can associate a probability density function (PDF) with the variable u. This PDF, P , is defined so that the probability that r < u < r+dr is P (r)dr, with r being the sample space variable of u. The sample space, over which u takes
I
V 1 !V 2
I 2
2
U 12 I 1 I 1
(a)
I
I
I 1
I 2
2
1
U 12 U 12 !
(b)
I
I
I 1
I 2
2
1
U 12
(c)
Figure 2.2: Schematics of various joint PDF’s of two scalar fluctuations. (a) ρ 12 = 0; (b) ρ 12 < 0 and ρ 12 > 0; (c) A case with ρ 12 = 0, which does not imply statistical independence.
2.1.3 Averaging with PDF
One of the most important uses of the PDF is that it can be used as an averaging tool. Averaging over the PDF is understood as averaging over the sample space from which the PDF has been compiled. The mean of u and of a function f (u) are defined by:
u =
uP (u)du (2.5)
f (u) =
f (u)P (u)du (2.6)
The variance of a variable is usually denoted by σ^2 and is defined by
σ^2 =
(u − u)^2 P (u)du (2.7)
= u^2 − 2 u u + u^2 = u^2 − u^2 (2.8)
The variance is a measure of the width of the PDF. In Reynolds averaging, we write u = u+u′, which then gives u^2 = u^2 + 2u′u + u′^2. After averaging,
u^2 = u^2 + 0 + u′^2 ⇔ u′^2 = u^2 − u^2 (2.9)
Comparison with Eq. (2.8) shows that the variance is the average of the fluctuations squared.
2.1.4 Correlation coefficient
For multivariate PDFs, we have similar results:
u =
uP (u, v)dudv ; v =
vP (u, v)dudv
σ^2 u =
(u − u)^2 P (u, v)dudv
uv =
uvP (u, v)dudv
u′v′^ =
(u − u)(v − v)P (u, v)dudv
= uv − u v − u v + u v = uv − u v (2.10)
Equation (2.10) provides a way by which the correlation between two random variables can be found easily experimentally (i.e. by building up the product uv, averaging, and then subtracting the product of the means). Finally, the correlation coefficient, ρuv is defined by
ρuv =
u′v′ σuσv
and takes values between −1 and 1. A positive correlation implies that when u′^ is positive (i.e. u is above its mean u), then it is likely that v′^ is also positive and vice versa. A negative correlation implies that it is likely that the fluctuations have different signs. A zero correlation sometimes is thought to imply that the random variables are independent, but this is not, strictly speaking, correct (Fig. 2.2c). It may happen that the joint PDF is symmetric (and hence the correlation coefficient is zero), but at the same time there may be a very good chance of knowing one of the fluctuations when we know the other. This point is very important when trying to draw physical conclusions from experimental or numerical data.
2.2 Turbulence Cascade, Scales and Eddies
2.2.1 Energy cascade
Most introductions to turbulence start from a statement that turbulence has a range of scales. A famous rhyme attributed to Richardson goes “Big whorls have little whorls that
σ Lturb
∂φ ∂x 2
Equations (2.14) and (2.15) can be used to provide the order of magnitude of the spatial gradients of mean quantities and relate them to the magnitudes of the turbulent quantities. We will use such estimates in Chapter 3 when we will examine the relative importance of the various terms in the governing equations.
2.2.4 Dissipation
A very important property of turbulent flows is the dissipation of kinetic energy per unit mass (units: m^2 /s^3 ), denoted by ε. Dissipation is a process that proceeds at a rate dictated by the large eddies, although it occurs at the smallest eddies. The rate at which kinetic energy is dissipated is equal to the rate at which kinetic energy cascades from the large to the small eddies. This rate is of the order of k/Tturb, which gives a very useful estimate for ε:
ε ≈
u^3 Lturb
Using the dissipation and the turbulent kinetic energy one can define a turbulent timescale and a lengthscale as
Tε =
k ε
Lε =
k^3 /^2 ε
which is done often in turbulence modelling. Note that these scales are somewhat different than the ones defined from Eqs. (2.13) and (2.16) because k = 3u^2 /2. The lengthscale defined from Eq. (2.18) is called the dissipation lengthscale and is equal to Lturb to within a constant^1. For the purposes of this course, we may use Lε and Tε as equivalent to Lturb and Tturb, but beware of the differences if more accurate work is necessary. Equation (2.16) is one of the key equations of turbulence and should never be forgotten. It has some very important implications. It shows that the magnitude of the viscosity does not really affect the dissipation, although it is by molecular processes that dissipation of kinetic energy to heat actually occurs. ε is independent of the Reynolds number of the flow, since it depends only on u and Lturb. However, the small-scale velocity and the lengthscale at which viscous effects become important depend on the kinematic viscosity ν and ε.
2.2.5 Reynolds number and Kolmogorov scales
The energy dissipation is independent of eddy scale, but the smallest motions will be affected by viscosity. Hence the smallest scales will be functions only of ε and ν. Kolmogorov used this
(^1) Later, we will see that we may define Lturb as the integral lengthscale L 11 , in which case experiment in
isotropic turbulence gives that Lε ≈ 2. 2 L 11.
fact to define the smallest length, time, and velocity scales possible in turbulence respectively as:
ηK =
ν^3 /ε
τK = (ν/ε)^1 /^2 (2.20) vK = (νε)^1 /^4 (2.21)
These equations can also be considered as dimensional necessities. From these definitions and the definition of the turbulent Reynolds number
Ret =
uLturb ν
we obtain the very useful:
ηK = LturbRe− t 3 /^4 (2.23) τK = TturbRe− t 1 /^2 (2.24) vK = uRe− t 1 /^4 (2.25)
These relations can be used to find the smallest scales of the motion, given the magnitude of the large-eddy scales. It can easily be seen that vK ηK /ν = 1, i.e. the flow at these small scales is viscous and hence below this scale, kinetic energy is being lost to heat by viscosity. Actually, viscosity effects are not quite negligible also at scales larger than ηK.
2.2.6 Intermediate scales
Since the energy flowing in the “cascade” is independent of the eddy scale, then we can argue that for any eddy of scale r having characteristic velocity v(r):
ε =
u^3 Lturb
v(r)^3 r
v K^3 ηK
This idea is explored further later that predicts how the turbulence energy is distributed in each eddy size. It can also be explored for simple estimates concerning phenomena (e.g. dispersion of a scalar patch) that occur at lengthscales between the large-eddy and the Kolmogorov scales.
2.2.7 What are eddies?
Eddies are difficult to define and there is no such thing as a “typical eddy”. Many quantities are used for their mathematical description. A picture often used in shown in Fig. 2.3: the Kolmogorov eddy is a vortex tube of thickness ηK with velocity vK (i.e. with angular velocity 1/τK ), spanning a region of length Lturb, with a characteristic undulation length equal to λ (λ lies in the range ηK < λ < Lturb and is called the Taylor microscale, to be
values that would be used in calculating the autocorrelation. In stationary turbulence (i.e. u′^2 is not changing with time), the autocorrelation depends only on τ. The autocorrelation coefficient is a very important quantity and is defined by
ρ(τ ) =
u′(t)u′(t + τ ) u′^2
The autocorrelation coefficient takes values between −1 and 1 and shows on average how related, or coherent, is the value of the velocity (or scalar) and t + τ with that at t. If ρ(τ ) is close to unity, this means that the fluctuation at t + τ is perfectly correlated, in a statistical sense, with its value at t, for any t. If it is zero, then the velocity at t + τ is statistically independent from the velocity at t. From the autocorrelation coefficient, we can define properly the integral timescale as
T (^) turb′ =
0
ρ(τ )dτ (2.28)
We have used the symbol T (^) turb′ in order to contrast this timescale from the eddy turnover time Tturb that we used previously. The reason will become apparent shortly. Therefore, the autocorrelation says something about the large scales.
2.3.2 Taylor temporal microscale
Another common use of ρ(τ ) is to give information about the small scales, and this is achieved by fitting a parabola at τ = 0 and measuring the time delay where this parabola crosses the τ axis (Fig. 2.4). This is defined from
∣ ∣ ∣ ∣
d^2 ρ dτ 2
τ =
λ^2 T
where λT is the Taylor microscale (for time). T (^) turb′ gives an estimate of time over which u(t) is correlated with itself, while the Taylor microscale provides information about the time derivative of u.
2.3.3 The convergence of averages
For short time delays, the two events are statistically correlated. Only for time delays τ > T (^) turb′ it is reasonable to argue that the two samples u(t) and u(t + τ ) are statistically independent. This is extremely important for experimentalists and modellers alike. There is no point in compiling averages over time series lasting only 1-2 T (^) turb′ , since the averages will simply not have converged. One needs long averaging times, not fast sampling rates (unless one needs to resolve the gradients, in which case one needs both). See Tennekes and Lumley (Ch. 6) [1] for more details. Here, we just quote a useful result that is often used in estimating the statistical uncertainty of averages. If UT is the true mean value of the random variable u and σ^2 is the true variance, then if we use N samples to estimate the mean Uest, or we are averaging only over a time T when the integral timescale is T (^) turb′ , then
|Uest − UT | UT
σ UT
N 1 /^2
σ UT
(T (^) turb′ T
Equation (2.30) gives the percentage error of our estimate, which must be quoted in both experimental and numerical studies of turbulence as as to quantify the statistical uncertainty of our data. Equation (2.30) says that signals with very large fluctuations relative to the mean (σ/UT = o(1)) require a great number of samples for the averages to converge. In contrast, for very small fluctuations (σ/UT ø 1), a smaller number of samples may give acceptable convergence.
τ t^ t
φ’
(a)
λT T τ
ρ(τ)
turb
(b)
Figure 2.4: (a) A schematic of a scalar time series and the time delay τ used to calculate the autocorrelation. (b) A typical autocorrelation coefficient and its use to calculate the integral timescale and the Taylor microscale.
2.3.4 Spatial autocorrelation and integral lengthscale
Similarly to temporal autocorrelations, we can define spatial correlations, where the two events are now at two different locations. These are called “two-point correlations” and quantify better the eddy size. For the velocity (a vector), we can define the correlation tensor as
Rij (r) =
u′ i(x)u′ j (x + r) √ u′ i^2 u′ j^2
where r is the vector joining the two points x and x + r. By homogeneous turbulence we mean that ∂/∂x is zero for all statistical moments.
Isotropic turbulence is defined as the turbulence where u′ 12 = u′ 22 = u′ 32 and u′ iu′ j = 0 (i 6 = j). For isotropic homogeneous turbulence, most of the components of the correlation tensor are equal and only a few components remain. In isotropic turbulence, we need only the longitu- dinal and the transverse correlations, given by
This equation provides a link between temporal and spatial correlations. It is valid only if there is a predominant mean flow in one direction and weak turbulence ( u/u 1 << 1, say less that 0.1). In this case, T (^) turb′ = Lturb/u. The eddy turnover time is defined as Tturb = Lturb/u. Note that the turbulent process of transferring energy from the large to the small scales proceeds according to Tturb, independent of u, while the temporal autocorrelation measured by an instrument fixed in space will be T (^) turb′. Hence, an estimate of the fastest frequency that our instrument must follow is u/ηK in order to resolve the smallest eddy as it is convected past our probe at the mean velocity. In isotropic turbulence,
ε ≡ 2 νs′ ij s′ ij = 15ν
∂u′ 1 ∂x 1
where s′ ij is the fluctuating strain rate equal to 12 (∂u′ i/∂xj + ∂u′ j /∂xi). We will discuss the definition of energy dissipation in more detail in Chapter 3. The spatial Taylor microscale λ is defined by
( ∂u′ 1 ∂x 1
u′^2 λ^2
The Taylor (length) microscale can also be visualized through the correlation coefficient for small separations (Fig. 2.5). With good enough instruments, i.e. from two closely-spaced hot wires, the velocity at two points can be measured and hence the quantity ∂u′ 1 /∂x 1 can
be calculated and then λ can be determined. Hence from the relation ε = 15νu′ 12 /λ^2 , ε can be estimated. If the Taylor hypothesis is valid (and often it is not very bad), then
ε =
15 ν u^21
∂u′ 1 ∂t
This equation has been used very often to estimate ε from single-point temporal measure- ments in turbulent flows.
2.3.6 Scalar scales
The same definitions for the temporal and spatial autocorrelations and two-point correlations can be applied to a scalar φ. Hence, we can define:
ρ(τ ) =
φ′(t)φ′(t + τ ) φ′^2
R(r) =
φ′(x)φ′(x + r) φ′^2
T (^) φ′ =
0
ρ(τ )dτ (2.41)
Lφ =
0
R(r)dr (2.42)
Tφ = Lφ/u (2.43)
The integral scalar timescales can be different from the velocity scales. Hence, Lφ 6 = Lturb and Tφ 6 = Tturb in general. However, if the velocity and scalar turbulence evolve together and have similar initial conditions we can argue that the scalar and the velocity scales are similar. Usually in thin shear flows (such as the jets, wakes and boundary layers we will examine in this course), the scalar length scale is similar to the velocity scale and this allows a simpler treatment.
2.4 Intermittency
Measurements of velocity or scalar signals in turbulent flows have revealed two particular features. In turbulent jets, wakes, and boundary layers, there is a clear distinction between the turbulence-bearing fluid (e.g. the jet or the boundary layer) and ambient fluid that is non-turbulent. This is called external intermittency and results in spiky scalar PDF’s. Internal intermittency has also been observed, which means that, even in a fully turbulent fluid, the regions that have high gradients (dui/dxj )^2 and (dφ/dxj )^2 occupy a small fraction of the volume. Figure 2.6 shows d′u/dt)^2 from a hot wire signal in a flow where Taylor’s hypothesis is valid (and therefore we can assume that du′/dt = udu′/dx). It is evident that the velocity gradient can obtain very high values for short periods of time, but this does not happen very often. The fraction of time when d′u/dt)^2 is very high decreases as Ret → ∞. This idea is qualitatively consistent with the eddy structure sketch in Fig. 2.3, which shows a small patch of intense vorticity (and hence high velocity gradients) in a large region of relatively quiescent fluid.
2.5 Spectra
2.5.1 Energy spectrum
There is a very useful technique that enhances our understanding of turbulent flows and this is based on Fourier analysis. This also brings our intuitive ideas of the energy cascade into a more solid ground.
Suppose we want to “distribute” the kinetic energy of the turbulence into eddy sizes, i.e. consider which eddy has what percentage of the total kinetic energy. This is physi- cally equivalent to decomposing our velocity fluctuations (e.g. Fig. 2.1) into its frequency components and measuring how much energy is contained in the various frequencies. After some mathematics (see any turbulence textbook), for isotropic turbulence we get that
k =
u′ iu′ i =
0
E(κ)dκ (2.44)
The physical interepretation of E(κ) is that it represents the kinetic energy (per unit mass) contained at wavenumber κ, so that the energy at κ is κE(κ). The wavenumber has units of m−^1. The energy spectrum E(κ) is large at low wavenumbers (large eddies) and low at high wavenumbers (small eddies). Equation (2.44) implies that the area under the E(κ) − κ curve represents all the turbulent kinetic energy. This provides a suitable way to normalise spectra. For example, Fig. 2.7 shows that the spectrum at low wavenumbers looks identical for all Ret, if E(κ) has been normalised by the large-scale quantities u and Lturb. The small scales, however, are shifted to higher wavenumbers (e.g. the small eddies become smaller) as Ret increases. Normalization with the small-scale quantities has the opposite effect. At high wavenumbers, viscous effects become important and the spectrum decays fast.
2.5.2 The inertial range -5/3 law
Experiments show that a characteristic feature of the energy spectrum is that there is a range of intermediate wavenumbers where the spectrum decreases as κ−^5 /^3. This is the inertial range and is found only at high Ret and there is ample evidence for it. This − 5 /3 law is a major result in turbulence theory. It can be derived from the following argument. Our usual assumptions concerning the cascade are that:
- The dissipation ε is independent of scale;
- viscosity is important only in the small scales.
These amount to saying that
E(κ) = f (κ, ε) (2.45)
The units of κ, E(κ), and ε are respectively m−^1 , m^3 s−^2 (which can be seen from Eq. 2.44), and m^2 s−^3. Hence, from Buckingham’s Theorem (3 quantities, 2 units), we get
E(κ) = Cκ−^5 /^3 ε^2 /^3 (2.46)
This is a dimensional necessity. The constant C is found from experiment and is of order one. Equation (2.46) is one of the simplest and yet deepest equations in turbulence. An alternative way to understand Eq. (2.46) is the following. Say that v(r) is the velocity at scale r. Our picture that the energy flowing in the cascade is dissipated only at the small scales implies an equality of dissipation across the inertial range. Since the energy is v^2 (r) = κE(κ) and r = 1/κ, using Eq. (2.46) gives that the lifetime of an eddy at
wavenumber κ must be proportional to ε−^1 /^3 κ−^2 /^3. This is equal to the turnover time of this eddy, defined by its size over its velocity, i.e. (1/κ)/[κE(κ)]^1 /^2. This shows that the eddy turnover time at any scale is roughly equal to eddy lifetime, which is an implicit assumption we’ve used before.
2.6 Exercises
- Derive a relationship between Ret and Reλ (Ret = uLturb/ν; Reλ = uλ/ν).
- The turbulent flow in a wind tunnel is isotropic and homogeneous. The mean velocity is u = 10 m/s, the characteristic turbulent intensity is u/u = 0.1 and the large-eddy lengthscale is Lturb = 0.01 m. Estimate the Kolmogorov scales ηK , vK , and τK , the Taylor microscale λ, ε, k, Ret, Reλ, Tturb, and T (^) turb′. What is the fastest frequency that a hot-wire probe will see? If we want 1000 statistically-independent samples, for how long should we be sampling?
- Assume that the turbulence in the atmosphere has a characteristic velocity u and large-eddy lengthscale Lturb. A patch of pollutant of characteristic size X is released. Assuming that ηK < X < Lturb and that dX/dt ≈ v(X), where v(X) is the velocity of eddies of size X, find how X increases with time.
- Show that vK ηK /ν = 1.