, the ensemble average is called an expected value and is denoted with angle brackets. With this terminology and notation, the mth-moment, 
, of the random variable u at location x and time t is defined as the ensemble average of um:12.3. Nomenclature and Statistics for Turbulent Flow
The dependent-field variables in a turbulent flow (velocity components, pressure, temperature, etc.) are commonly analyzed and described using definitions and nomenclature borrowed from the theory of stochastic processes and random variables even though fluid-dynamic turbulence is not entirely random. Thus, the characteristics of turbulent-flow field variables are commonly specified in terms of their statistics or moments. In particular, a turbulent field quantity, ϑ ˜ 
, is commonly separated into its first moment, ϑ ¯ 
, and its fluctuations, ϑ ≡ ϑ ˜ − ϑ ¯ 
, which have zero mean. This separation is known as the Reynolds decomposition and is further described and utilized in Section 12.5.
, is commonly separated into its first moment, , and its fluctuations, , which have zero mean. This separation is known as the Reynolds decomposition and is further described and utilized in Section 12.5.To define moments precisely, specific terminology is needed. A collection of independent realizations of a random variable, obtained under identical conditions, is called an ensemble. The ordinary arithmetic average over the collection is called an ensemble average and is denoted herein by an over bar. When the number N of realizations in the ensemble is large, N → ∞ u m ¯
, the ensemble average is called an expected value and is denoted with angle brackets. With this terminology and notation, the mth-moment, , of the random variable u at location x and time t is defined as the ensemble average of um:
(12.1)
where u(x,t:n) is nth the realization in the ensemble. The limit N → ∞ 
can only be taken formally in theoretical analysis, so when dealing with measurements, u m ¯ 
is commonly used in place of 〈 u m 〉 
and good experimental design ensures that N is large enough for reliable determination of the first few moments of u. Thus, the over-bar notation for ensemble average is favored in the remainder of this chapter. Collectively, the moments for integer values of m are known as the statistics of u(x,t).
can only be taken formally in theoretical analysis, so when dealing with measurements, is commonly used in place of and good experimental design ensures that N is large enough for reliable determination of the first few moments of u. Thus, the over-bar notation for ensemble average is favored in the remainder of this chapter. Collectively, the moments for integer values of m are known as the statistics of u(x,t).Under certain circumstances, ensemble averaging is not necessary for moment estimation. When u is stationary in time, its statistics do not depend on time, and u m ¯ 
at x can be reliably estimated from time averaging:
at x can be reliably estimated from time averaging:
(12.2)
when Δt is large enough. Time averages are relevant for turbulent flows that persist with the same boundary conditions for long periods of time, an example being the turbulent boundary-layer flow on the hull of a long-range ship that traverses a calm sea at constant speed. Example time histories of temporally stationary and non-stationary processes are shown in Figure 12.2. When u is homogeneous or stationary in space, its statistics do not depend on location, and u m ¯ 
at time t can be reliably estimated from spatial averaging in a volume V:
at time t can be reliably estimated from spatial averaging in a volume V:
Figure 12.2 Sample time series indicating temporally stationary (a) and non-stationary processes (b). The time series in (b) clearly shows that the average value of u decreases with time compared to the time series in (a).
(12.3)
when V is large enough and defined appropriately. This type of average is often relevant in confined turbulent flows subject to externally imposed temporal variations, an example being the in-cylinder swirling and tumbling gas flow driven by piston motion and valves flows in an internal combustion piston engine.
Throughout this chapter, all moments denoted by over bars are ensemble averages determined from (12.1), unless otherwise specified. Equations (12.2) and (12.3) are provided here because they are commonly used to convert turbulent flow measurements into moment values. In particular, (12.2) or (12.3) are used in atmospheric and oceanic field measurements because ongoing natural phenomena like weather or the slow meandering of ocean currents make it practically impossible to precisely repeat field observations under identical circumstances. For such measurements, a judicious selection of Δt or V is necessary; they should be long or large enough for reliable moment estimation but small enough so that the resulting statistics are only weakly influenced by ongoing natural variations not of fluid mechanical origin.
Before defining and describing specific moments, several important properties of the process of ensemble averaging defined by (12.1) must be mentioned. First, ensemble averaging commutes with differentiation, that is, the application order of these two operators can be interchanged:
Similarly, ensemble averaging commutes with addition, multiplication by a constant, time integration, spatial differentiation, and spatial integration. Thus the following are all true:
(12.4, 12.5, 12.6)
(12.7, 12.8, 12.9)
where v is another random variable; a, b, m, and A are all constants; and dx represents a general spatial increment. In particular, (12.5) with m = 0 implies A ¯ = A 
, so if A = u ¯ 
then u ¯ ¯ = u ¯ 
; the ensemble average of an average is just the average. However, the ensemble average of a product of random variables is not necessarily the product of the ensemble averages. In general,
, so if then ; the ensemble average of an average is just the average. However, the ensemble average of a product of random variables is not necessarily the product of the ensemble averages. In general,
The simplest statistic of a random variable u is its first moment, mean, or average, u ¯ 
. From (12.1) with m = 1, u ¯ 
is:
. From (12.1) with m = 1, is:
(12.10)
In general, u ¯ 
may depend on both space and time, and is obtained by summing the N separate realizations of the ensemble, u(x,t:n) for 1 ≤ n ≤ N, at time t and location x, and then dividing the sum by N. A graphical depiction of ensemble averaging, as specified by (12.10), is shown in Figure 12.3 for time-series measurements recorded at the same point x in space. The left panel of the figure shows four members, u(x,t:n) for 1 ≤ n ≤ 4, of the ensemble. Here the average value of u decreases with increasing time. Time records such as these might represent atmospheric temperature measurements during the first few hours after sunset on different days, or they might represent a component of the flow velocity from the cylinder of a compressor in the first 10 or 20 milliseconds after an exhaust valve opens. The right panel of Figure 12.3 shows the ensemble average u ( x , t ) ¯ 
obtained from the first two, four, and eight members of the ensemble. The solid smooth curve in the lower right panel of Figure 12.3 is the expected value that would be obtained from ensemble averaging in the limit N → ∞ 
. The dashed curve is a time average computed from only the fourth member of the ensemble using (12.2) with m = 1 and Δt equal to one-tenth of the total time displayed for each time history. Figure 12.3 clearly shows the primary effect of averaging is to suppress fluctuations since they become less prominent as N increases and are absent from the expected value. In addition, it shows that differences between an ensemble average of many realizations and a finite-duration temporal average of a single realization may be small, even when the flow is not stationary in time.
may depend on both space and time, and is obtained by summing the N separate realizations of the ensemble, u(x,t:n) for 1 ≤ n ≤ N, at time t and location x, and then dividing the sum by N. A graphical depiction of ensemble averaging, as specified by (12.10), is shown in Figure 12.3 for time-series measurements recorded at the same point x in space. The left panel of the figure shows four members, u(x,t:n) for 1 ≤ n ≤ 4, of the ensemble. Here the average value of u decreases with increasing time. Time records such as these might represent atmospheric temperature measurements during the first few hours after sunset on different days, or they might represent a component of the flow velocity from the cylinder of a compressor in the first 10 or 20 milliseconds after an exhaust valve opens. The right panel of Figure 12.3 shows the ensemble average obtained from the first two, four, and eight members of the ensemble. The solid smooth curve in the lower right panel of Figure 12.3 is the expected value that would be obtained from ensemble averaging in the limit . The dashed curve is a time average computed from only the fourth member of the ensemble using (12.2) with m = 1 and Δt equal to one-tenth of the total time displayed for each time history. Figure 12.3 clearly shows the primary effect of averaging is to suppress fluctuations since they become less prominent as N increases and are absent from the expected value. In addition, it shows that differences between an ensemble average of many realizations and a finite-duration temporal average of a single realization may be small, even when the flow is not stationary in time.
Figure 12.3 Illustration of ensemble and temporal averaging. The left panel shows four members of an ensemble of time series for the decaying random variable u. In all four cases, the fluctuations are different but the decreasing trend with increasing t is clearly apparent in each. The right panel shows averages of two, four, and eight members of the ensemble in the upper three plots. As the sample number N increases, fluctuations in the ensemble average decreases. The lowest plot on the right shows the N → ∞ 
curve – this is the expected value of u(t) – and a simple sliding time average of the n = 4 curve where the duration of the time average is one-tenth of the time period shown. In this case, time and ensemble averaging produce nearly the same curve.
curve – this is the expected value of u(t) – and a simple sliding time average of the n = 4 curve where the duration of the time average is one-tenth of the time period shown. In this case, time and ensemble averaging produce nearly the same curve.Although useful and important in many situations, the average or first moment alone does not directly provide information about turbulent fluctuations. Such information is commonly reported in terms of one or more higher-order central moments defined by:
(12.11)
where in practice u ( x , t ) ¯ 
often replaces 〈 u ( x , t ) 〉 
. The central moments primarily carry information about the fluctuations since (12.11) explicitly shows that the mean is removed from each ensemble member. The first central moment is zero by definition. The next three have special names: ( u − u ¯ ) 2 ¯ 
is the variance of u, ( u − u ¯ ) 3 ¯ 
is the skewness of u, and ( u − u ¯ ) 4 ¯ 
is the kurtosis of u. In addition, the square root of the variance is known as the standard deviation and is frequently denoted by the subscript rms for root-mean-square: ( u − u ¯ ) 2 ¯ = u r m s . 
In the study of turbulence, a field variable's first moment and variance are most important.
often replaces . The central moments primarily carry information about the fluctuations since (12.11) explicitly shows that the mean is removed from each ensemble member. The first central moment is zero by definition. The next three have special names: is the variance of u, is the skewness of u, and is the kurtosis of u. In addition, the square root of the variance is known as the standard deviation and is frequently denoted by the subscript rms for root-mean-square: In the study of turbulence, a field variable's first moment and variance are most important.Example 12.3
Compute the time average of the function u ( t ) = A e − t / τ + B cos ( ω t ) 
using (12.2). Presuming this function is meant to represent a turbulent field variable with zero-mean fluctuations, Bcos(ωt), superimposed on a decaying time-dependent average, A e − t / τ , 
what condition on Δt leads to an accurate recovery of the decaying average? And, what condition on Δt leads to suppression of the fluctuations?
using (12.2). Presuming this function is meant to represent a turbulent field variable with zero-mean fluctuations, Bcos(ωt), superimposed on a decaying time-dependent average, what condition on Δt leads to an accurate recovery of the decaying average? And, what condition on Δt leads to suppression of the fluctuations?Solution
Start by directly substituting the given function into (12.2):
and evaluating the integral:
This can be simplified to find:
In the limit Δ t → 0 
, both factors in [,]-braces go to unity and the original function is recovered. Thus, the condition for properly determining the decaying average is Δt ≪ τ; the averaging interval Δt must be short compared to the time scale for decay, τ. However, to suppress the contribution of the fluctuations represented by the second term, its coefficient must be small. This occurs when ωΔt ≫ 1 which implies the averaging interval must be many fluctuation time periods long. Therefore, a proper averaging interval should satisfy: 1 ≪ ωΔt ≪ ωτ, but such a choice for Δt is not possible unless ωτ ≫ 1.
, both factors in [,]-braces go to unity and the original function is recovered. Thus, the condition for properly determining the decaying average is Δt ≪ τ; the averaging interval Δt must be short compared to the time scale for decay, τ. However, to suppress the contribution of the fluctuations represented by the second term, its coefficient must be small. This occurs when ωΔt ≫ 1 which implies the averaging interval must be many fluctuation time periods long. Therefore, a proper averaging interval should satisfy: 1 ≪ ωΔt ≪ ωτ, but such a choice for Δt is not possible unless ωτ ≫ 1.