Environmental Fluid Mechanics - Chapter 13 pot

13 Topics in Stratified Flow 13.1 BUOYANCY AND STABILITY CONSIDERATIONS Nearly all natural surface water bodies are stratified at least part of the time. This means there are density variations, usually in the vertical direction. Horizontal variations also may exist, but not in steady state unless there are other forces such as Coriolis effects present to balance the resulting pressure differences (see Chap. 9). Density variations exist most commonly because of temperature and/or salinity gradients. Salinity is the main contributor to density in the oceans and in some inland lakes such as the Great Salt Lake in Utah or the Dead Sea in Israel, though these water bodies also may have temperature gradients. In freshwater lakes, temperature stratification is most important. In fact, from a water quality point of view, there is usually great interest in modeling the temperature structure of water bodies. Most biological and chemical reactions depend on temperature, and fish choose habitats based partly on this parameter. As discussed in Chap. 12, gas transfer across the air/water interface also depends on temperature. A closely related parameter to density is buoyancy,definedas b D g  0    0 D g 0 13.1.1 where  is the density of a fluid parcel and  0 is the reference density. Buoy- ancy is also known as reduced gravity, g 0 , and is defined so that a fluid parcel tends to rise when its buoyancy is positive (i.e., its density is less than the reference value) and a particle with higher buoyancy has a greater tendency to rise. Buoyancy may be treated like other state variables, such as temperature or concentration, and a conservation equation can be defined, ∂b ∂t C  V Ð  rb D  rÐk b  rb Csource/sink 13.1.2 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. where k b is diffusivity for buoyancy and the source/sink terms are defined depending on which component is contributing to the buoyancy. For example, a solar heating term might generate buoyancy in a temperature-stratified system (Sect. 12.2), while evaporation might cause (negative) buoyancy at the surface, due either to cooling (heat loss due to evaporation) or to increased salinity in the case of saline water. Usually, an equation of state linking density or buoy- ancy to these other properties is needed in addition to the general equations of motion in order to define these systems. 13.1.1 Equation of State Density is related to temperature and salinity and possibly other properties of a particular system through an equation of state. A well-known example of such an equation is the perfect gas law, which relates density, pressure, and temperature through the gas constant. Similarly, a general equation of state may be formulated for water systems as  D T, C, p 13.1.3 where T D temperature, C D concentration of dissolved species, and p D pressure. Since the main stratifying agent for most natural systems, in terms of dissolved species, is salinity S, this will be used in place of C. Also, except in the deep oceans or the atmosphere, the incompressible assumption implies that density should not be a function of pressure. Thus  D T, S 13.1.4 A number of expressions have been proposed to define this relationship, usually based on high-order polynomial fits to tabulated values of density as a function of T and S. Relations have been proposed for simple sodium chloride solutions and also to actual seawater solutions. In general the dependence on temperature has been found to be approximately parabolic, with a maximum at 4 ° C (Fig. 13.1). However, the temperature of the density maximum changes with increasing salinity, decreasing to about 0 ° C for highly saline systems. To a first-order approximation, density is linearly dependent on salinity over much of the normal range of interest. The dependence of density on T and S is expressed through values of the thermal and saline expansion coefficients, ˛ and ˇ, respectively, where ˛ D 1  0 ∂ ∂T 13.1.5 and ˇ D 1  0 ∂ ∂S 13.1.6 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 13.1 Variation of freshwater density with temperature. The negative sign in the definition for ˛ is meant so that, at least for T>4 ° C, the coefficient has a positive value. Note that ˛ changes sign for temperatures less than the temperature of the density maximum. In general, both ˛ and ˇ are functions of T and S, but approximate values are ˛ ¾ D 2 ð 10 4 ° C 1 and ˇ ¾ D 7.5 ð 10 3 S 1 ,whereS is in weight percent. It is clear from these values that salinity has a much greater effect on density than does temperature. The equation of state is written to express density in terms of ˛ and ˇ, and deviations of T and S from standard or reference values as  D  0 1 ˛T C ˇS 13.1.7 where T D T  T 0 , S D S  S 0 ,  D  0 when T D T 0 and S D S 0 ,and T 0 and S 0 are the reference values for temperature and salinity, respectively. Normally, T 0 D 4 ° CandS 0 D 0 (%). Note that S is usually expressed in units of weight percent, parts per thousand, or mg/L. Typical seawater has S ¾ D 3.5%, 35 ppt (o/oo), or 35,000 ppm. When ˛ and ˇ are taken as constants, the resulting equation is called a linear equation of state. A constant value for ˇ is usually a good approximation, but in order to account for the parabolic nature of the temperature dependence, ˛ should be a function of T.Asimple expression that gives reasonable results over much of the range of normally Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. occurring values for T and S (except near freezing) is  D  0 [1  0.00663T 4 2 C 7.615S] 13.1.8 where T is in ° CandS is in weight percent. For freshwater bodies, with S D 0, this equation is reasonable even near freezing. When S is high, however, greater than about 5%, higher order equations should be used to estimate . 13.1.2 Gravitational Stability When density differences exist in a fluid system, an important considera- tion is that of stability. In Chap. 10 the concept of convective transport was introduced, where it was noted that convection is the result of a gravitation- ally unstable condition. This is simply saying that lighter, more buoyant fluid should tend to rise and that the system would be stable only when heavier fluid underlies lighter fluid. Buoyancy instabilities give rise to convective motions, which tend to mix the fluid system vertically. This is demonstrated mathemat- ically by applying a perturbation analysis to a system in which there is no motion initially (u D v D w D 0), with a density stratification z, as illus- trated in Fig. 13.2. A fluid particle is considered with initial position at z D 0, which is an arbitrary point within the fluid. Figure 13.2 Fluid particle at z D 0, in fluid with ambient stratification given by z D  r (reference distribution). Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. The governing equation for this problem is  0 D 1  rÐp C  g ) ∂p ∂z Dg 13.1.9 This is simply the momentum equation for the vertical direction, which reduces to the hydrostatic result for the case of zero flow. The reference density,  r , must satisfy this equation. An equation of state also should be considered to relate density to other properties of the system, as noted above. The fluid is assumed to be incompressible (note that an incompressible fluid does not have to have the same density everywhere). If the fluid particle of Fig. 13.2 is displaced vertically from its equilibrium location by an amount z, there will be a buoyancy force acting on the particle due to the difference in density between the particle and its new surroundings. The resulting force acting on the particle, per unit volume, is [g p   r ], where  p is the fluid particle density. Applying Newton’s law,  F D m  a ) F 8 D m 8 a ) g r   p  D  p Rz13.1.10 where m is the mass of the fluid particle, 8 is its volume, and the vector notation is dropped with the understanding that the force balance is in the vertical (z) direction. The double dots over z in the last part of this equation indicate a second derivative with time (i.e., acceleration). Taylor series expansions are now defined for  r and  p ,intermsofa reference value  0 :  r D  0 C z d r dz C [Oz 2 ] 13.1.11  p D  0 C z d p dz C [Oz 2 ] 13.1.12 where the last term in both of these expressions indicates that the neglected or truncated terms in the series approximations are of order Oz 2  (i.e., second order). These terms will be neglected in the following. Assuming there is no heat transfer, the relationship between pressure and density is given by the adiabatic relationship, dp D c 2 d,wherec D sonic velocity. Making these substitutions, along with the hydrostatic pressure result (13.1.9), into Eq. (13.1.10), we have d p dz ¾ D d 0 dz D 1 c 2 dp dz D  0 g c 2 13.1.13 and  p Rz D g   0 C z d r dz   0 C z  0 g c 2  [COz 2 ] D gz   0 g c 2 C d r dz  13.1.14 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. The Brunt–Vaisala or buoyancy frequency, N, is defined in terms of the square root of the negative of the term in parentheses in this last expression. Note that the density gradient is assumed to be negative, as it must be for a gravitationally stable water column; N is undefined for an unstable density distribution. However, the sonic velocity c is normally large, and the first term in parentheses on the right-hand side of Eq. (13.1.4) can be neglected under most conditions. This is equivalent to neglecting the small compressibility of the fluid parcel (resulting in a small change in density) due to the change in ambient pressure at the perturbed location. In general, N is defined for most applications by N 2 D g  0 d r dz 13.1.15 and Eq. (13.1.14) then becomes  p Rz DN 2 z p 13.1.16 where it has been assumed that  p ¾ D  0 . The final differential equation is Rz CN 2 z D 0 13.1.17 which is an equation of simple harmonic motion. Equation (13.1.17) has solutions of the form z / e iNt N 2 > 0 13.1.18a z / e jNjt N 2 < 0 13.1.18b z / e 0 N 2 D 0 13.1.18c These three solutions correspond to stable, unstable, and neutral conditions, respectively. In other words, N 2 > 0 implies a gravitationally stable condi- tion, with density increasing with depth. When N 2 < 0, heavier fluid overlays lighter fluid, which is an unstable condition leading to convection (in fact, N is undefined in this situation, as noted above). A neutrally stable condition is one where there is no acceleration since there is no net gravitational force acting on the fluid particle when it is displaced from its original equilibrium position, and particle position remains constant. For the stable situation, substituting the Euler formula, e ši D cos  š i sin 13.1.19 it is seen that oscillatory motions are expected, with amplitude depending on the magnitude of the original displacement of the fluid particle. These oscillations decay over time due to viscous effects, which have not been considered here but could be included in the equation of motion for the fluid Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. particle, Eq. (13.1.17), if desired. For the unstable case, the particle position grows exponentially with time. Thus any perturbation of particle position in an unstable environment will lead to large-scale convective motions. 13.2 INTERNAL WAVES We now consider wave motions that are possible in a stratified fluid. Internal waves can propagate along the interface between fluid layers of different densi- ties (note that surface waves, as discussed in Chap. 8, propagate along the air/water interface, which is an extreme example of fluids of two different densities) or, more generally, at an angle to the horizontal through a density stratified fluid, with N 2 > 0. Consider a stratified fluid with density and pressure fields given by   x,t D  0 C  0 z C 00 x, y, t 13.2.1 p  x,t D p 0 C p 0 z Cp 00 x, y, t 13.2.2 where x and y are the horizontal coordinates, subscript 0 indicates a reference value, a single prime denotes a function of z only (vertical position) and the double prime indicates a function of horizontal position and time. Normally, it can be assumed that the magnitude of  0 is much greater than the magnitude of  0 , which in turn is much greater than the magnitude of  00 (one or two orders of magnitude difference between each component). Since we are dealing primarily with water, incompressibility dictates that the density following a fluid particle is constant, or D Dt D ∂ ∂t C  u Ðr D 0 13.2.3 As shown in Chap. 2, this leads to the usual continuity equation for an incom- pressible fluid, rÐ  u D ∂u j ∂x j D 0 13.2.4 The general momentum equation is (neglecting Coriolis terms), Du k Dt D ∂u k ∂t C u j ∂u k ∂x j D 1  ∂p ∂x k C g k C r 2 u k 13.2.5 For the analysis of stratified fluids it is convenient to consider a reference state of zero motion. The reference density is  r D  0 C  0 z (see Fig. 13.2). The pressure is given similarly as p r D p 0 C p 0 z. Substituting this definition for Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.  r into Eq. (13.2.3) results in D Dt D D r Dt C D 00 Dt D ∂ r ∂t C u j ∂ r ∂x j C D 00 Dt D 0 ) D 00 Dt Dw d r dz 13.2.6 since  r is a function of z only. The momentum equation for the reference state is simply the hydrostatic result, 0 D ∂p r ∂x k C  r g k 13.2.7 Subtracting Eq. (13.2.7) from Eq. (13.2.5) results in  Du k Dt D ∂p  p r  ∂x k C    r g k C r 2 u k 13.2.8 Then, using the definitions for p r and  r , along with the approximation that 1/ ¾ D 1/ 0  (this is the Boussinesq approximation discussed in Sect. 2–7), Du k Dt D 1  0 ∂p 00 ∂x k C  00  0 g k C r 2 u k 13.2.9 This is the governing equation for momentum, though in the following we also will neglect viscous effects (high Reynolds number assumption). A wave equation is derived by writing the momentum equations sepa- rately for the horizontal and vertical directions: ∂u h ∂t C 1  0 ∂p 00 ∂x h Du j ∂u h ∂x j 13.2.10 ∂w ∂t C  00  0 g Du j ∂w ∂x j 13.2.11 where u h denotes a horizontal velocity component, i.e., h takes values of 1 or 2, for the x or y direction, respectively. The fact that p 00 6D fz has also been taken into account in writing Eq. (13.2.11). Taking the derivative of Eq. (13.2.10) with respect to z and subtracting the derivative of Eq. (13.2.11) with respect to x h gives ∂ ∂t  ∂u h ∂z  ∂w ∂x h   ∂ ∂x h   00  0 g  D ∂ ∂x h  u j ∂w ∂x j   ∂ ∂z  u j ∂u h ∂x j  13.2.12 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. This last result is then differentiated with respect to t and the two component equations (for h D 1or2)arethen ∂ 2 ∂t 2  ∂u ∂z  ∂w ∂x   ∂ 2 ∂t ∂x   00  0 g  D ∂ 2 ∂t ∂x  u j ∂w ∂x j   ∂ 2 ∂t ∂z  u j ∂u ∂x j  13.2.13 ∂ 2 ∂t 2  ∂ ∂z  ∂w ∂y   ∂ 2 ∂t ∂y   00  0 g  D ∂ 2 ∂t ∂y  u j ∂w ∂x j   ∂ 2 ∂t ∂z  u j ∂ ∂x j  13.2.14 Differentiating the first of these with respect to x and differentiating the second with respect to y and adding the results gives ∂ 2 ∂t 2  ∂ 2 u ∂x ∂z C ∂ 2 v ∂y ∂z  ∂ 2 w ∂x 2  ∂ 2 w ∂y 2   g  0  ∂ ∂t r 2 h  00   D ∂ ∂t  r 2 h  u j ∂w ∂x j   ∂ 3 ∂t ∂x h ∂z  u j ∂u h ∂x j  13.2.15 where r 2 h D ∂ 2 ∂x 2 C ∂ 2 ∂y 2 13.2.16 is a two-dimensional horizontal Laplacian operator and index notation is used for subscript h. The first term on the left-hand side is simplified using the continuity Eq. (13.2.4): ∂u ∂x C ∂ v ∂y D ∂w ∂z ) ∂ 2 u ∂x ∂z C ∂ 2 v ∂y ∂z D ∂ 2 w ∂z 2 13.2.17 The second term on the left-hand side also is rewritten using Eq. (13.2.6): g  0  ∂ ∂t r 2 h  00   D g  0 r 2 h  ∂ 00 ∂t  D g  0 r 2 h  D 00 Dt  u j ∂ 00 ∂x j  D g  0 r 2 h  w ∂ r ∂z  u j ∂ 00 ∂x j  D g  0   ∂ r ∂z r 2 h w r 2 h  u j ∂ 00 ∂x j  13.2.18 Substituting these last two results into Eq. (13.2.15), rearranging, and using the definition of the buoyancy frequency (13.1.15) gives ∂ 2 ∂t 2 r 2 w CN 2 r 2 h w D ∂ 3 ∂t ∂x h ∂z  u j ∂u h ∂x j  r 2 h  g  0 u j ∂ 00 ∂x j C ∂ ∂t  u j ∂w ∂x j  13.2.19 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. The right-hand side is generally negligible compared with the other terms in the equation, since it involves all nonlinear terms. A possible exception to this is when there is strong mean shear, in which case the velocity gradients may be important. The simplified final equation is ∂ 2 ∂t 2 r 2 w CN 2 r 2 h w ¾ D 0 13.2.20 which is a wave equation similar to Eq. (13.1.17). It is beyond the scope of the present text to provide a full discussion of the solutions to this equation (or the more general Eq. 13.2.19) and the resulting behavior of the fluid motions that it describes. There are many books that cover this material in depth, and the present analysis is restricted to a discussion of some properties of linear internal waves and their relationship to surface waves. First, however, we consider the lowest mode solutions, which correspond to horizontal propagation. 13.2.1 Lowest Mode Solutions By assuming wavelike disturbances, w D Wz exp[ik 1 x Ck 2 y t] 13.2.21 where k 1 and k 2 are wave numbers for the x and y directions, respectively, and  is frequency, Eq. (13.2.20) becomes d 2 W dz 2 C  N 2   2  2  k 2 h W D 0 13.2.22 where k 2 h D k 2 1 C k 2 2 . This is a wave equation for W, which expresses wave propagation in the horizontal direction. Boundary conditions must be applied at the surface and at the bottom. The free surface is defined by Áx, y, t, as sketched in Fig. 13.3. At the surface, z D Á, the dynamic boundary condition is p r Á Cp 00 x h ,Á,tD p atm D 013.2.23 and the kinematic boundary condition is DÁ Dt D w13.2.24 However, application of boundary conditions is problematic at z D Á,since Á itself is an unknown, obtained as part of the solution. For a first-order solution, it is more convenient to write the surface boundary conditions at z D Á D 0 and to account for variations between this level and the actual Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. [...]... differential operator on the left-hand side of Eq (13. 3.15) may be written as ∂ ∂ CU ∂t ∂x D ik U c 13. 3.16 Then, combining Eqs (13. 3.14) and (13. 3.15) gives w1 jzD0C D ik U1 O c Á D C1 O Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 13. 3.17a w2 jzD0 D ik U2 O c Á D C2 O 13. 3.17b Since these must be equal, a relationship between C1 and C2 is obtained, U2 U1 C2 D C1 c c 13. 3.18 A second matching... ∂t ∂x u2 0 D 1 2 g ik Á O 13. 3.22 This result may be written, using Eq (13. 3.16) as 1 ik U1 c u1 0 C 2 ik U2 c u2 0 D 1 2 g ik Á O 13. 3.23 Using Eq (13. 3.10), along with Eq (13. 3.14) to evaluate dw0 /dz at the interface (z D 0), we find u1 0 D i dw1 0 D k dz u2 0 D iC1 i dw1 0 D iC2 k dz 13. 3.24 Substituting these expressions for u1 0 and u2 0 into Eq (13. 3.23), along with Eq (13. 3.17), assuming C1 6D... along with Eqs (13. 3.49) and (13. 4.5), ∂uj D0 ∂xj ∂u 1 ∂p0 D C r2 u ∂t 0 ∂x 1 ∂p0 ∂v D C r2 v ∂t 0 ∂y 0 ∂w 1 ∂p0 g C r2 w D ∂t 0 ∂z 0 T ∂T0 w D kT r2 T0 ∂t h Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 13. 4.6 13. 4.7 13. 4.8 13. 4.9 13. 4.10 ∂S0 ∂t w S D kS r2 S0 h 13. 4.11 where (u, v, w) has been substituted for (u1 , u2 , u3 ) We now take the derivative of Eq (13. 4.7) with respect to z and... ∂x ∂z ∂x ∂y ∂z D g 0 ∂2 0 ∂2 0 C 2 ∂x 2 ∂y 2 ∂ w ∂y 2 13. 4.12 Making a substitution as in Eq (13. 2.17) for the velocity gradients, this becomes ∂ ∂t r2 r2 w D gr2 h ˛T0 C ˇS0 13. 4 .13 where Eq (13. 4.4) has been used to substitute for density Nondimensional equations are developed by introducing scaling quantities into Eqs (13. 4.3), (13. 4.10), and (13. 4.11): h for length, h2 / for time, kT /h for velocity,... Rights Reserved 13. 3.46 Figure 13. 8 System definition sketch for analysis of free convection p D ps C p 0 ∂ps ˛T D g 0 1C z ∂z h 13. 3.47 13. 3.48 where ps is hydrostatic pressure and primes indicate perturbation quantities (i.e., small variations from the initial condition) The temperature equation (13. 3.39), upon substituting from Eq (13. 3.46), becomes ∂T0 ∂T0 C uj ∂t ∂xj w T D kT r2 T0 h 13. 3.49 where... substituting Eq (13. 2.26) gives ∂p00 ∂t 0 gw D0 13. 2.27 Taking the time derivative of the linearized vertical momentum equation (from Eq 13. 2.11), we obtain g ∂p00 ∂2 w D C ∂t2 0 ∂t 1 ∂2 p00 0 ∂t∂z 13. 2.28 and the linearized mass conservation equation (from Eq 13. 2.6) is ∂p00 D ∂t w d r dz Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 13. 2.29 Substituting this last result into Eq (13. 2.28), along... lower layers, respectively (Fig 13. 6) Following the preceding analysis, vertical velocity w0 is assumed to be described by an equation of the form of Eq (13. 3.8) Within each layer, N D 0 and U is constant, so Eq (13. 3.12) reduces to d2 w O 2 dz k2w D 0 O Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 13. 3 .13 Figure 13. 6 Definition sketch for stability analysis of two-layer system (note the similarity... Wz D 0>z> h h>z> H 13. 2.37 and from Eq (13. 2.35), sinh[kh H h ] sinh kh h C1 D C2 13. 2.38 while Eq (13. 2.36) implies 2 D gkh  fkh υ C coth kh h C coth[kh H h ]g 1 13. 2.39 0 This last result is known as a dispersion relationship, which relates kh and If the interface is relatively thick, or kh υ ! 1, the wavelengths of the disturbance are short compared with υ, and 2 ¾ g D υ 0 13. 2.40 For the other... interfacial region is irrotational, a velocity potential ϕ can be defined so that wjzD h ∂ϕ Dw ∂z Given the assumed form for w, ϕ is assumed as r2 ϕ D 0 ϕ D  z exp[i k1 x C k2 y 13. 2.46 t] 13. 2.47 Then, using Eq (13. 2.37) for the region above the interface, d D W D C1 sinh kh z ) C1 sinh kh h dz iA D iA ) C1 D sinh kh h 13. 2.48 so that D iA cosh kh z kh sinh kh h 13. 2.49 At the surface, the linearized horizontal... p0 C p 0 ∂p0 D ∂x ∂x C 0 13. 3.3 and 0 D ∂ w0 ∂w0 ∂ w0 C U C u0 C w0 ∂t ∂x ∂z ∂ p0 C p 0 g 0C 0 ∂z C 0 13. 3.4 Upon carrying out the multiplications on the right-hand sides and linearizing with respect to the perturbation quantities, we obtain 0 ∂u0 C ∂t 0U ∂u0 C ∂x ∂w0 C ∂t 0U 0w 0 dU dz ¾ D ∂p0 ∂x 13. 3.5 and 0 ∂w0 ¾ D ∂x ∂p0 ∂z g 0 C 0 13. 3.6 Subtracting the derivative of Eq (13. 3.5) with respect to . (13. 1.9), into Eq. (13. 1.10), we have d p dz ¾ D d 0 dz D 1 c 2 dp dz D  0 g c 2 13. 1 .13 and  p Rz D g   0 C z d r dz   0 C z  0 g c 2  [COz 2 ] D gz   0 g c 2 C d r dz  13. 1.14 Copyright. Eq. 13. 2.6) is ∂p 00 ∂t Dw d r dz 13. 2.29 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Substituting this last result into Eq. (13. 2.28), along with the boundary condi- tion (13. 2.27),. Eq. (13. 3.8). Within each layer, N D 0andU is constant, so Eq. (13. 3.12) reduces to d 2 Ow dz 2  k 2 Ow D 0 13. 3 .13 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 13. 6 Definition