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Turbulence is an important subject. In fact, most flows encountered in engineering are turbulent and not laminar. This term paper discusses the velocity distribution in turbulent flow of an incompressible fluid. Foremost, the characteristics of turbulent flows were mentioned because of the difficulty to give a precise definition of turbulence. Also, the experimental comparison of turbulent and laminar flows and a few empiricisms that have been proposed for turbulent momentum flux were considered.
Most flows occurring in nature and engineering applications are turbulent not laminar. The water currents below the surface of oceans, the flow of rivers and canals are turbulent; the Gulf Stream is a turb- ulent wall-jet kind of flow. Also, the boundary layers in the earth's atmosphere (except possibly in very stable condition); the jet streams in upper troposphere are turbulent; cumulus clouds are in turbulent motion. Most combustion processes involve turbulence and often even depend on it; the flow of natural gas and oil pipes is turbulent. Chemical engineers use turbulence to mix, homogenize and transport fluid mixtures and to accelerate chemical reaction rates in liquids or gases.
1.1 Characteristics of turbulent flow
It is quite difficult to give a precise definition of turbulence. The best description is with a set of characteristics of turbulent flows:
Irregularity: One characteristic of turbulent flows is their irregularity or randomness. A full deterministic approach is very difficult. Turbulent flows are usually described statistically. Turbulent flows are always chaotic. But not all chaotic flows are turbulent.
Diffusivity: The diffusivity of turbulence causes rapid mixing and increased rates of momentum, heat, and mass transfer. A flow that looks random but does not exhibit the spreading of velocity fluctuations through the surrounding fluid is not turbulent. If a flow is chaotic, but does not exhibit spreading of velocity fluctuations through the surrounding the fluid, it is surely not turbulent.
Dissipation: Turbulent flows are dissipative. Kinetic energy gets converted into heat due to viscous shear stresses. Turbulent flows die out quickly when no energy is supplied. Random motions that have insignificant viscous losses, such as random sound waves, are not turbulent.
Three-dimensional vorticity fluctuations: Turbulent flows are rotational; that is, they have non-zero vorticity. Mechanisms such as the stretching of three-dimensional vortices play a key role in turbulence.
Large Reynolds number: Turbulent flows always occur at high Reynolds numbers. Turbulence often originates as a result of instability of laminar flows, if the Reynolds number becomes too large. The instabilities are related to the interaction of viscous terms and nonlinear inertia terms in the equations of motion. They are caused by the complex interaction between the viscous terms and the inertia terms in the momentum equations.
It is crucial to know that turbulence is not a feature of a fluid but of fluid flows. Most of the dynamics of turbulence is the same in all fluids, whether they are liquids or gases.
Comparison of Laminar and Turbulent flow
Foremost to any theoretical details discussion about turbulence. It is important to summarize the differences between laminar and turbulent flows in several simple systems.
Circular Tubes: For a steady, fully developed, laminar flow in a circular tube of radius R the velocity distribution and the average velocity are given by
and (2.0-1, 2)
and that the pressure drop and mass flow rate w are linearly related:
Considering turbulent flow on the other hand, the velocity is fluctuating with time chaotically at each point in the tube. The "time-smoothed velocity" can be measure at each point with a Pitot tube. This type of instrument is not sensitive to rapid velocity fluctuations, but senses the velocity averaged over several seconds. The time-smoothed velocity has a z-component represented by and its shape and average value will be given very roughly by
and (2.0-4, 5)
This power expression for the velocity distribution is too crude to give a realistic velocity derivative at the wall. The laminar and turbulent velocity profiles are compared in Fig. 2.1-1.
Fig. 2.1-1. Qualitative comparison of laminar and turbulent velocity profiles.
Over the same range of Reynolds numbers the mass rate of flow and the pressure drop are no longer proportional but are related approximately by
The laminar-turbulent transition in circular pipes normally occurs at a critical Reynolds number of roughly 2100, although this number may be higher if extreme care is taken to eliminate vibrations in the system.
Noncircular Tubes: For developed laminar flow in the triangular duct the fluid particles move rectilinearly in the z direction, parallel to the walls of the duct. By contrast, in turbulent flow there is superposed on the time-smoothed flow in the z direction (the prim a y pow) a time-smoothed motion in the xy-plane (the secondary flow). The secondary flow is much weaker than the primary flow and manifests itself as a set of six vortices arranged in a symmetric pattern around the duct axis (see Fig. 2.1-2b). Other noncircular tubes also exhibit secondary flows.
Fig. 2.1-2. Sketch showing the secondary flow patterns for turbulent flow in a tube of triangular cross section [H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, 7th edition (1979), p. 613]
Flat Plate: It was found that for the laminar flow around a flat plate, wetted on both sides, the solution of the boundary layer equations gave the drag force expression as:
where is the Reynolds number for a plate of length L, the plate width is W and the approach velocity of the fluid is . Also, for turbulent flow, the dependence on the geometrical and physical properti- es is quite different:
Thus the force is proportional tothe power of the approach velocity for laminar flow, but to the power for turbulent flow. The stronger dependence on the approach velocity reflects the extra energy needed to maintain the irregular eddy motions in the fluid.
Time-Smoothed Equations of Change for Incompressible Fluids
In turbulent flow the velocity is fluctuating in a chaotic fashion as shown in Fig 3.0-1(a). The fluct- uations are irregular deviations from a mean value. The actual velocity can be regarded as the sum of the mean value and the fluctuation mean. For example, for the z-component of the velocity we write
. (3.0- 1)
which is sometimes called the Reynolds decomposition. The mean value is obtained from by making a time average over a large number of fluctuations (3.0- 2)
the period being long enough to give a smooth averaged function. For the system at hand, the quantity which we call the time-smoothed velocity, is independent of time, but of course depends on position. When the time-smoothed velocity does not depend on time, we speak of steadily driven turbulent flow. The same comments we have made for velocity can also be made for pressure. Next we consider turbulent flow in a tube with a time-dependent pressure gradient. For such a flow one can define time-smoothed quantities as above, but one has to understand that the period to must be small with respect to the changes in the pressure gradient, but still large with respect to the periods of fluctuations. For such a situation the time-smoothed velocity and the actual velocity are illustrated in Fig. 3.0-2(b). In similar manner the relations also hold for x and y directions and these fluctuations can also occur in the x and y direction.
Fig. 3.0-2. Sketch showing the velocity component as well as its time-smoothed value and its fluctuation in turbulent flow (a) for "steadily driven turbulent flow" in which does not depend on time, and (b) for asituation in which does depend on time.
Turbulent Shear or Reynolds stresses
In the fluid flowing in turbulent flow shear forces occur wherever there is a velocity gradient across a shear plane and these are much larger than those occurring in laminar flow. The velocity fluctuations in Eq. (3.0- 1) gives rise to turbulent shear stresses. We start by writing the equations of continuity and motion with replaced by its equivalent and by its equivalent .The equation of continuity is then , and we write the x-component of the equation of motion
+ (3.1- 3)
+ (3.1- 4)
Now we use the fact that the time-averaged value of the fluctuating velocities is zero , and that the time averaged product is not zero. Then Eqs. (3.1- 1) and (3.1- 2) become
+ (3.1- 6)
By comparing these two time -smoothed equations with Eqs. (3.1- 1) and (3.1- 3) we can conclude that the time -smoothed everywhere replace the instantaneous values. However, in Eq. (3.1- 6) new terms arise in the set of brackets which are related to turbulent velocity fluctuations. For convenience we use the notation
, (3.1- 7)
These are the components of the turbulent momentum flux and are called Reynolds stresses.
Empirical Expressions for the Turbulent Momentum Flux
The equations derived for turbulent flow must be solved to obtain velocity profiles. To do this, more simplifications must be made before the expressions for the Reynolds stresses can be evaluated. A number of semi-empirical equations have been used; the eddy-diffusivity model of Boussinesq is one early attempt to evaluate these stresses and Prandtl mixing length expression.
The Eddy Viscosity of Boussinesq: By analogy with Newton's law of viscosity, one may write for a turbulent shear flow
in which is the turbulent viscosity (often called the eddy viscosity),is a strong function of position and the intensity of turbulence. For two kinds of turbulent flows (i.e., flows along surfaces and flows in jets and wakes), special expressions for are available:
( i) Wall turbulence: (3.2- 2)
This expression is valid only very near the wall. It is of considerable importance in the theory of turbulent heat and mass transfer at fluid-solid interface.
( ii) Free turbulence: (3.2- 3)
in which K, is a dimensionless coefficient to be determined experimentally, b is the width of the mixing zone at a downstream distance z, and the quantity in parentheses represents the maximum difference in the z-component of the time-smoothed velocities at that distance z. Prandt1 found Eq. 3.2-3 to be a useful empiricism for jets and wakes.
The Mixing Length of Prandtl: In his mixing-model Prandtl developed an expression for momentum transfer in a turbulent fluid by assuming that eddies move around in a fluid very much as molecules move around in a low-density gas. The eddies move a distance called the mixing length L before they lose their identity. This kind of reasoning led Prandtl to the following relation: (3.2- 3)
If the mixing length were a universal constant, Eq. 5.4-4 would be very attractive, but in fact has been found to be a function of position. Prandtl proposed the following expressions for :
( i) Wall turbulence: (3.2- 4)
( ii) Free turbulence: (3.2- 5)
in which and are constants. A result similar to Eq. 3.2-3 was obtained by Taylor by his "vorticity transport theory" some years prior to Prandtl's proposal.
Turbulent Flow in Ducts
The case of turbulent flows through pipes was investigated very thoroughly in the past because of its great practical importance. Moreover, the result arrived at are important not only for pipe flow; they also contribute to the extension of our fundamental knowledge of turbulent flow in general. In order, to give some impressions about the Reynolds stresses, a short discussion of experimental measurements for turbulent flow in rectangular ducts are considered in Figs. 4.0-1 and 2 are shown some experimental measurements of the time-smoothed quantities , and for the flow in the direction in a rectangular duct. In Fig. 4.0-1 note that quite close to the wall is about 13% of the time-smoothed centerline velocity , whereas is about 5%. This means that, near the wall, the velocity fluctuations in the flow direction are appreciably greater than those in the transverse direction. Near the center of the duct, the two fluctuation amplitudes are nearly equal and we say that the turbulence is nearly isotropic there.
In Fig. 4.0-2 the turbulent shear stress; is compared with the total shear stress across the duct. It is evident that the turbulent contribution is the more important over most of the cross section and that the viscous contribution is important only in the vicinity of the wall.
Fig. 4.0-1. Measurements of H. Reichardt Fig. 4.0-2. Measurements of Reichardt (see Fig. 4.0-1)
[Naturwissensckaften, 404 (1938), Zeits. for the quantity in a rectangular duct .Note that
f. angew. Math. u. Mech., 13,177-180 this quantity only near the duct wall.
(1933), 18,358-361 (1938)] for the turbulent
flow of air in a rectangular duct with
Here the quantities
and are shown .
In chemical engineering, turbulent flows play an important role in the successful design of processes, e.g. in the form of heat and mass transfer under turbulent pipe flow conditions. An important characteristic of turbulence is its ability to transport and mix fluid much more effectively than comparable laminar flow, and turbulence greatly enhances the rates of these processes. No doubt, that the biggest part of nature is turbulent and to be turbulent is natural.