flow, sediment, and nutrient transport in a riparian … · web view2008/04/06 · title flow,...

Turbulence: Mean and Fluctuating Components of Motion

byEka Oktariyanto Nugroho

Basic Equation 6 Eka O. N.

6.1. THE DEFINITION OF MEAN MOTION AND MEAN FORCES

6.1.1. Characteristics of Mean Motion vs Actual Motion

In the previous chapters, theory was sometimes illustrated by examples in which the real motion was actually turbulent, despite the fact that we were dealing only with an ideal fluid.In these examples, it was implied but not specified that only the average values of the velocity and the pressure were considered. In a turbulent motion, the true velocity and pressure vary in a disorderly manner. In fact, a turbulent motion is always unsteady, since at a given point the velocity changes continuously in a very irregular way. It is also nonuniform, since the velocity changes from point to point at a given time, and rotational, since the friction forces, proportional to , are important. These characteristics are true as far as the actual motion is concerned. However, by splitting the motion into mean and fluctuating components, the average motion may often be considered steady, uniform (in a pipe), or irrotational (over a weir). Now this method has to be justified and the differences between the motion of an ideal fluid or a viscous flow, and a mean turbulent motion have to be further considered. This is the purpose of this section.

6.1.2. Validity of the Navier-Stokes Equation for Turbulent Motion

Equalities between the inertia forces and the applied forces on an elementary fluid particle are valid even if the motion is turbulent. Hence, the basic Navier-Stokes equations and continuity relationships are also theoretically valid in the study of turbulent motion. It has been seen that it is sometimes possible to calculate a laminar solution where the boundary conditions are simple. It is also possible to determine theoretically the stability of the solution that is to determine whether a small disturbance will increase or be damped out by friction. However, a fully deterministic approach is no longer possible in the case of turbulent motion, because of the random nature of these turbulent fluctuations.On the other hand, in engineering practice, it is not always necessary to know the exact fine structure of the flow. Only the average values and the overall and statistical effects of turbulent fluctuations have to be studied.

6.1.3. Definitions of the Mean Values in a Turbulent Flow

In a turbulent motion, as in the case of a viscous flow, velocity and pressure have to be known as functions of the space coordinates and time.

The instantaneous velocity at a fixed point is the vectorial sum of the mean velocity with respect to time (referring to the basic primary movement) and the fluctuation velocity which varies rapidly with time both in intensity and direction. This can be expressed by the relationships

where, by definition,

and

or we can define the turbulent time-averaged variables as

where and T = an appropriate time period(a time interval)And .

Turbulence: Mean and Fluctuating Components of Motion Page - 40


T is large compared to the time scale of the turbulent fluctuations.T is small compared to the time scale of variation of the turbulent time averaged flow.We assume that the behavior is ergodic. (i.e. the statistics are stationary for period T)

Similarly, the instantaneous pressure is the sum of the mean pressure and a fluctuation term such that where

and

Similarly, the instantaneous components of turbulent time averaged are defined as follows:Table 6. 1 Turbulent time-averaged variables.

Real velocity

Mean velocity

Fluctuation velocity

= +

= +

= +

= +

= +

Figure 6. 1 Sketch definition of time average turbulent.Hence, turbulent motion may be considered as the superposition of a mean motion and a fluctuating and disorderly motion, random in nature, which obeys statistical laws.

6.1.4. Steady and Unsteady Mean Turbulent Flows

The mean value is defined for intervals of time T which are large compared to the time scale of turbulent fluctuations but small compared to the time scale of the mean motion.If, for example, one considers the oscillation of water in a tunnel between a surge tank and a reservoir, the instantaneous velocity at a fixed point may vary quickly because of the turbulence. The average velocity defined for a relatively short interval of time varies also with respect to time, but its change is slow.The real motion is always unsteady because of turbulence and in this case, the mean motion is also unsteady (Fig. 6.2). In the following discussion, a motion is called unsteady only if the mean value of the velocity varies. The interval of time, which permits a realistic definition of the mean motion, is relative to the frequency of turbulent fluctuations. It varies with the phenomenon to be studied.



Figure 6. 2 The steadiness of a turbulent flow is defined by the mean velocity only.

6.1.5. Mean Forces

Since the real value of the inertia forces is always equal to the sum of the real values of the applied forces in any kind of motion (laminar or turbulent), the mean value of the inertia forces with respect to time is equal to the mean value of the applied forces with respect to time. This may be expressed as shown in Equation 6-1. Since

one also has:

(6-1)

Equation 6-l is expressed mathematically, along the OX axis, as (see Section 5.1)

Or, using the and notations and the rotational coefficients (see sections 5.1)

Similar equations are found for the OY and OZ axes. Now each of these mean forces has to be expressed as a function of the mean values and fluctuating values of the velocity and the pressure. For this purpose, one considers:

l. The constant forces Gravity force2. The linear forces Pressure force, linear function of

Local inertia force, linear function of

Friction force, linear function of

3. The quadratic force Convective inertia, function of or product of two components of

velocity:



6.2. CALCULATION OF THE MEAN FORCES

The mean forces are calculated as a function of the mean values of velocities and pressure. For these calculations, it is assumed that the order of mathematical operations has no effect on the final result. In particular, integration during an interval of time T and derivatives with respect to time or space can be interchanged.

6.2.1. The Constant Force

The gravity force depends only on the density of the elementary particle. The fluctuations of pressure are too small to have a significant effect on the density. Hence, the gravity force is the same for laminar and turbulent motion. The mean value of the gravity force is equal to this constant gravity force. Mathematically, this may be expressed

Since is a constant with respect to time, then

6.2.2. Linear Forces

The mean value of the local inertia force, may be obtained by considering the mean value of any of its components. For example,

Introduce , where is a fluctuation term, into this equation.

The mean value is given by: .

Since is determined as a constant over time T:

thus

Hence, the mean value of the local inertia force with respect to time is equal to the inertia force caused by the change of value of the mean velocity alone.

Similarly averaging the pressure forces yields, for example:



The viscous force has a mean value which may be calculated by considering, for example, one of

the second order terms, such as . Averaging this term leads also to

The mean viscous force is equal to the viscous force due to the mean velocity alone, and is mathematically expressed in the same way as the actual motion. All the linear forces involved in the mean motion are mathematically written in the same way for both mean turbulent flow and actual motion, turbulent or laminar.

6.2.3. The Quadratic Forces

Consider the component . Squaring and averaging leads successively to

This is a result of the following intermediate steps. Since is a constant in the interval of time T:

Since

Since u' may be positive or negative but is always positive and its mean value is different from zero:

, similarly, the mean value of is and the mean values of

and are zero.

Now, consider the mean value of any term of convective inertia, such as . One has successively.

Considering each of these terms independently one obtains, since and are constant with respect to time,

Introducing these values yield



, similarly, it is found that and so on.

Hence, the mean value of a convective inertia force with respect to time is equal to the sum of the convective inertia caused by the mean velocity and the mean convective inertia caused by the turbulent fluctuations. As far as the mean value of the velocity alone is concerned, the convective inertia terms have the same mathematical form as for the case of a laminar motion.

6.3. THE CONTINUITY RELATIONSHIP

In the simple case of an incompressible fluid, the continuity relationship is written (see Section 2.2):

This relationship, expressed as a function of the mean components of velocity and turbulent fluctuations, becomes

Or

The averaging process, applied to , gives and applied to , gives

Then the continuity relationship for the mean motion becomes (see 6.2)

Consequently

The mathematical form of the continuity relationship is the same for the mean motion as for the actual motion. In vector notation, the turbulent time averaged continuity equation is:

Where

6.4. THE CHARACTERISTICS OF THE MEAN MOTION OF A TURBULENT FLOW

Insofar as the mean velocity and the mean pressure alone are concerned, the basic momentum equation and the continuity relationship have exactly the same mathematical form as the corresponding equations for the actual motion. However, other forces exist and have to be added. These new forces are caused by the convective inertia of the turbulent fluctuations. If these "new" forces may be neglected, or as long as only the forces which are functions of the mean velocity and mean pressure are dealt with, the solutions of problems concerning turbulent motion have the same mathematical form as the solutions given by the Navier-Stokes equations. For example, a mean motion which is steady and irrotational and for which the viscous forces are neglected obeys the Bernoulli equation:



as found in Section 5.1 where the velocity and the pressure now designate the average values. When applying this equation, the assumptions must be made relative to the mean motion; i.e., the mean motion must be steady, irrotational, and without viscous friction (despite the fact that the actual turbulent motion is always unsteady, rotational, and dissipative).

In practice the fluctuations of pressure are very small by comparison with the real pressure ,

so that . On the other hand, the viscous forces caused by the mean motion are generally small in comparison with the other forces, in particular with the convective inertia forces caused by the turbulent fluctuations. The viscous forces can often be neglected except, for example, in a laminar boundary sub layer between a smooth wall and a turbulent boundary layer. Now the effects of the convective fluctuating forces on the mean motion have to be studied. Then a relationship between the value of the mean velocity and the fluctuating velocity has to be established. Indeed since another unknown has been added, another relationship is necessary.

6.5. REYNOLDS EQUATIONS

6.5.1. Purpose of the Reynolds Equations

Expressing each force in the Navier-Stokes equation as a function of the mean values

and the fluctuating values and averaging, leads to the Reynolds equation. The Reynolds equation is the form of the Newton or momentum equation for turbulent motion.Since each of the mean forces has been calculated in the previous sections, it is possible to obtain directly the Reynolds equations. To do this, the sum of these mean forces is equated to zero. Recall that each force has the same mathematical form as in the Navier-Stokes equation, but it is expressed as a function of the mean values of velocity or pressure. However, additional convective inertia forces exist, caused by the fluctuating terms. For example, the mean value of the quadratic

inertia term, ), is

Hence, the momentum equation valid for the average motion in the OX direction may be written directly as Equation 6-2. (Since the calculation method is identical in the OY and OZ directions, only the momentum equation along the OX axis is studied)

(6-2)



Rewrite again, equation (5-2) and dividing by , equation yields:

(5-2)

Consider the X-direction Navier-Stokes equation and substituting the expanded variables:

Expanding and time averaging the entire equation over period T

Typical time averaging of products of time averaged variable terms:



Typical time averaging of products of time averaged variable terms and turbulent fluctuating components:

Time averaging of products of turbulent fluctuating components can not be simplified:

This simplifies the time averaged X-direction Navier-Stokes equation to:

By adding the continuity equation of the fluctuating components multiplied by u’ within the time averaging integral (by adding zero to the integral):

This leads to:

Finally we arrive at the final X-direction equation:

Grouping the viscous terms and turbulent time averaged fluctuations:

(6.3)Similarly for the Y and Z-directions turbulent time averaged Navier-Stokes equations are:



These equations are referred to as the Reynolds Equations. They are very similar to the Navier-Stokes equations except that:

- variables are turbulent time averaged quantities- There are now convective inertia forces caused by turbulent fluctuations

There are now two mechanisms of momentum transfer built into these equations:

- type terms are the Viscous Stresses and represent the averaged effect of molecular

motions. These terms are necessary since we are not directly simulating momentum transfer via molecular level collisions.

- type terms are the Turbulent Reynolds Stresses and represent the averaged effect of momentum transfer due to turbulent fluctuations. These terms are necessary when using turbulent time averaged variables since we are not directly simulating momentum transfer via turbulent fluctuations.

Note that the effective stress is

The Reynolds Equations greatly simplify turbulent flow computations in terms of:- Often allow a reduction of the number and complexity of the governing equations.- Simplify the time and space variability of the variables, allowing us to focus on much larger

spatial and temporal scales.

The Reynolds Equations include 6 new unknown terms: (We

can easily show that )

- We must introduce additional constitutive relationships to solve for these unknowns.- While the constitutive relationships for molecular level averaging was fluid and fluid

condition (temperature and pressure) dependent, the constitutive relationships for turbulent flow are both fluid and flow dependent.

- The simplest closure models are similar to a Newtonian stress-flow relationship. These may work well for very specific problems. An example of a simple turbulence closure model is the Boussinesq model for turbulence:

where = turbulent viscosity which is a flow dependent quantity

In vector notation, the Reynolds Equations are:



6.5.2. Reynolds Stresses

The convective inertia caused by the fluctuating velocity components is given in Section 6.2.3 as

The value of was shown to be zero by the continuity relationship in

Section 6.3. Thus, this expression can be added to the convective inertia without changing the value.When these two expressions are added and grouped in pairs, the result is:

This can be written as

Now, by introducing these terms (and two similar terms obtained for the OY and OZ directions) into the general momentum equation, the Reynolds equations (Equation 6-4) are obtained.

(6-4)

It is seen, indeed, that these Reynolds equations are very similar to the Navier-Stokes equations shown previously. The difference is in the convective inertia forces caused by the turbulent fluctuations and in the fact that the other forces are expressed as functions of the mean value of the velocity or pressure. The turbulent fluctuation forces are called “Reynolds stresses“.

6.5.3. Value of the Lame Components in a Turbulent Motion

The applied forces have been expressed independently of their physical nature, as is shown in Section 5.1.3. It is recalled, for example, that the applied forces along the OX axis are expressed by

The averaging process applied to these terms (which are either constant, such as X, or linear) gives for the applied forces:

Introducing this above expression in the Reynolds equation yields



From this equation it is easily deduced that the fluctuation terms may be considered as external forces which are added to the other forces defined by normal forces and shear stresses . Hence, these new external forces to be dealt with are:

Normal force:

Shear stress:

and so on. These new total external forces may also be defined by a tensor of rank two similar to the first tensor defined in Section 5-5.2. In practice the viscous forces caused by the mean velocity are very often negligible in turbulent flow in comparison with the other forces and particularly in comparison with the shear stresses caused by the fluctuation terms , and .

6.5.4. Correlation Coefficients and Isotropic Turbulence

By definition, in isotropic turbulence the mean value of any function of the fluctuating velocity components and their space derivatives is unaltered by a change in the axes of reference. In particular:

,

It is evident that isotropy introduces a great simplification in the calculations. However, because of the boundaries, the turbulence is not isotropic and the products may differ

from each other. There exists a correlation between and , and , and and , defined by the coefficients:

These coefficients are equal to zero in the case of isotropic turbulence. Since the convective inertia forces caused by the fluctuation terms are functions of , they may be expressed directly as functions of the coefficients of correlation which are dimensionless.


flow, sediment, and nutrient transport in a riparian … · web view2008/04/06 · title flow,...

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