Hydrodynamic Background
(2.5.46) In the present text the turbulent closure modelling is entirely limited to
the simple concept of an eddy viscosity. This kind of closure is also termed a simple algebraic closure model.
Advanced turbulence modelling
It is important to realize, however, that far more advanced closure models are available. These models require numerical computations and
42 Introduction t o nearshore hydrodynamics
are based on developing evolution equations for the turbulent quantities.
The first such quantity modelled is usually the turbulent kinetic energy k . The equation for is based on the energy equation for the turbulent flow.
is often modelled along with the dissipation of in a so called -
but this is only one type of turbulent closure model. Such advanced models of course have the advantage of being able to represent more realistically more complex flows. In the - and are used to determine a generalized (time and space varying) eddy viscosity by the expression
(2.5.47)
where is an emppirical constant. This is then normally used to determine the turbulent stresses by means of by which the closure of the equations is achieved.
However, all closure models have the same problem: in the evolution equations for each turbulent quantity such as or will always emerge other, more complicated turbulent quantitites such as triple correlations between turbulent fluctuations, that need to be estimated on the basis of physical reasoning, experimental results etc. Thus the closure problem is always just transferred t o other unknowns.
The spectrum of modern turbulence methods also includes the Large Eddy Simulation (LES) approach in which the larger eddies in the flow are included in the numerical results as part of the mean flow, and only the small scale turbulence is covered by empirical model equations. Modern computers are also becoming so powerful that Direct Numerical Simulation (DNS) by direct solution of the (viscous) Navier-Stokes equations, where all turbulent scales are computed, can give answers to small scale flow problems.
White (1991) gives a description of the - model concept, and Wilcox (1998) gives an extensive recent overview of a wider selection of turbulence techniques. The review of turbulence Reynolds-stress closure models by Speziale (1991) still remains useful in spite of the rapid developments in this area, and there is a large paper by Vreman et al. (1997) covering the LES method, to mention a few.
For more information reference is made to recent literature.
2.6 Energy in a flow 43
A
h udt
2.6 Energy flux in a flow
To analyze the energy flux in a flow across an arbitrary surface, we first consider a small element of the surface, see Fig. 2.6.1.
Fig. 2.6.1
section.
Definition sketch for analysis of the energy flux through a vertical
The energy flux through this element has two contributions:
(i) the actual flow of potential and kinetic energy carried by the water (ii) the work done by the pressure on the moving particles.
particles passing through the section.
The energy density (energy per unit volume) is therefore given by
+
1+ +
=+
12 2 (2.6.1)
where the first term is the potential energy relative to some (arbitrarily chosen) horizontal reference level, the second the kinetic energy.
Thus, the instantaneous energy flux through a surface element from transport of potential and kinetic energy is, per unit time
where
(2.6.2)
= (2.6.3)
is the velocity component normal to the surface element.
44 Introduction to nearshore hydrodynamics
The work done by the pressure is, per unit time
= (2.6.4)
so that the total energy flux through is
(2.6.5)
and we can determine the total energy flux by integrating over the entire surface
In most of our applications the surface considered is a plane vertical section so that = d z times the unit width and is the same for all z (see Figure 2.6.1). This implies that = n,, where index refers t o a vector in the y)-plane. Since we assume a plane surface here we can bring the constant normal vector n, outside the integral. The total instantaneous energy flux through such a vertical section therefore becomes
At this point, however, we have not yet assumed that the velocity vector has the same direction at all points over the section. In fact if we are considering a flow that is a combination of waves (that may be plane so that has the same direction over the vertical) and depth varying currents then obviously the total velocity will vary in direction over depth.
The total integral will then be a vector in a direction that depends on that depth variation, given by
valid for an arbitrary flow sending energy through a plane vertical surface extending from the bottom to the instantaneous free surface. The flux through the plane surface with normal vector n, is then
2.7 Appendix: Tensor notation 45
In vector notation this can be written
For completeness the x, y components of this are
(2.6.10)
(2.6.11) (2.6.12)
2.7 Appendix: Tensor notation
Tensor notation (or “index notation”) utilizes the fact that many equa- tions look the same in all three coordinate directions. Thus instead of writing out the x, y and equations, we use the symbols or j as indexes.
Each of those can then be x, y or
Fig. 2.7.1 Tensor-coordinate system.
Example: can be or w depending on whether j corresponds to x, y or
In addition, it is customary in tensor notation to talk about
instead of z . This i s illustrated in Fig. 2.7.1 showing the Cartesian coordinate system normally used in tensor notation.
46 Introduction t o nearshore hydrodynamics
For that reason, we also normally say that i (or is 1, 2, instead of Finally, tensor notation becomes really elegant because of the summation rule: If the same index is repeated as index within the same term, it means the term should be considered the sum of all the three terms we get by letting the repeated index be 1, 2 and 3. Thus we have as examples
= 722
+
733 (2.7.1)ii) - -
...
222) -
+ +
= -
+- +-
ax: ax:
Note that ii) is the same as
(2.7.2)
(2.7.3)
(2.7.4)
(2.7.5) which is the divergence of Similarly iw) expresses the Laplacian operator (= in some texts) (2.7.6) Kronecker
In tensor notation we also meet the so called Kronecker It is defined as
1 for i =
= 0 for
1 0 0
(2.7.7)
Hence is really the unit matrix.
Essentially, when occurs in a term in an equation, it has the effect of changing i to j (or j to i) in that term.
Example 1
(2.7.8)
8.8 References - Chapter 2 47
because by the summation rule
=
+ +
(2.7.9)However by the definition of only the term which has equal to the first index is 1, the others are 0. So
= 2 = (2.7.10)
= 1 =
= 3 =
(or which makes no difference because of the summation rule) and (2.7.12)
2.8 References
-
Chapter 2 Useful references on fluid mechanics:Aris, Rutherford (1962). Vectors, tensors and the basic equations of fluid Batchelor, G.K. (1968). An Introduction to Fluid Dynamics, Cambridge Currie, I.G. (1974). Fundamental mechanics of fluids. New Hinze (1975). Turbulence, Hill.
Kundu, P.K. (1990, 2001). Fluid Mechanics. Academic Press, New York.
Mei, C.C. (1983). The Applied Dynamics of Ocean Surface Waves, Wiley R. (1984). Incompressible flow. Wiley, New York.
Prandtl, L. (1952). Essentials of fluid dynamics. Hafner Publishing Comp., New York.
Speziale, C. G. (1991) Analytical methods for the development of Reynolds-stress closures in turbulence. Ann. Rev. Fluid Mech.
,
Tennekes, H. and J.L. Lumley (1972). A First Course in Turbulence, The mechanics. Prentice Hall, Englewood Cliffs, NJ.
Univ. Press.
York.
Interscience.
23, 107-137.
MIT Press.
48 Introduction t o nearshore hydrodynamics
Vreman, B., G. Bernard, and H. Kuerten (1997). Large eddy simulation White, F. M. (1974, 1991). Viscous fluid flow.
Wilcox, D. C. (1998). Turbulence modeling for CFD. DCW Industries, Yuan, (1967). Foundations of fluid mechanics, Prentice Hall,
of the turbulent mixing layer. J. Fluid Mech., 339, 357-390.
Inc., La Calif 91011.