Numerical modeling of the generation of internal ... - Wiley Online Library

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2012; 68:451–466 Published online 26 January 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2513

Numerical modeling of the generation of internal waves by uniform stratified flow over a thin vertical barrier H. Houcine1 , Yu. D. Chashechkin2 , P. Fraunié3, ∗, † , H. J. S. Fernando4 , A. Gharbi1 and T. Lili1 1 Laboratoire

de Mécanique des Fluides, Faculté des Sciences de Tunis, Université El Manar, 2092 Tunis, Tunisie of Fluid Mechanics, Institute for Problems in Mechanics of the RAS 101/1 prospect Vernadskogo, Moscow 119526, Russia 3 Laboratoire de Sondages Electromagnétiques de l’Environnement Terrestre, Université du Sud Toulon Var and Centre National de la Recherche Scientifique, BP 20136, F83957 La Garde, France 4 Department of Civil Engineering and Geological Sciences, University of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, IN 46556, U.S.A.

2 Laboratory

SUMMARY Numerical simulations of two-dimensional stratified flow past an obstacle (thin vertical strip) were performed at relatively low Reynolds numbers. A finite differences solver was adopted to simultaneously solve Navier–Stokes equations together with transport equations for salinity (stratifying agent), and the standard Smagorinsky turbulent closure scheme was called in whenever necessary to account for turbulence. The emphases were on the evaluation of code for unsteady stratified flow applications as well as identification of transient and steady internal-wave processes during flow past obstacles. Simulations were compared with laboratory experiments, where observations were made using a high resolution Schlieren technique and conductivity probes. Blocking was observed upstream of the obstacle, surrounded by nearzero frequency internal waves, the phase lines of which joined those of lee waves through a transition zone in the proximity of the obstacle. This pattern was preceded by initial transients of the starting flow in which propagating internal waves played a dominant role. Confluence of isopycnals passing over/under the obstacle in the wake led to interesting flow phenomena, including the radiation of internal waves. The numerical simulations were in good agreement with observations, except that some phenomena could not be captured due to resolution issues of either numerical or experimental techniques. The efficacy of the code in point for stratified flow calculations with relevance to the atmosphere and oceans was confirmed. Copyright 䉷 2011 John Wiley & Sons, Ltd. Received 9 September 2008; Revised 21 October 2010; Accepted 14 November 2010 KEY WORDS:

stratified flows; wakes; internal waves; computational fluid dynamics; large eddy simulations; comparison with experiment

1. INTRODUCTION Density stratification is ubiquitous in environmental flows and technological flows. In the oceans, it is mainly created by the non-uniformity of salinity and temperature whereas in the atmosphere the temperature, moisture, and pressure inhomogeneities are responsible for stratification. Density changes lead to unique and interesting set of phenomena, in particular, internal and lee waves, jet-like flow structures, thin interfaces with high density and velocity gradients, and anisotropic turbulence due to inhibition of vertical mixing by stratification. Internal waves are particularly interesting, wherein they effectively transport momentum but not the mass, and their breaking ∗ Correspondence

† E-mail:

to: P. Fraunié, LSEET, Université du Sud Toulon Var, BP 2032 F 83857 La Garde cedex, France. [email protected]

Copyright 䉷 2011 John Wiley & Sons, Ltd.

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causes turbulent patches that may decay to form fossil turbulence or sustain and grow into vertically thin but horizontally vast regions of property transport and high refractive-index fluctuations [1, 2]. In the atmosphere, these are the regions of Clear Air Turbulence and in the oceans they are called blinis [3]. Internal waves are generated by various mechanisms, for example, during the collapse of mixed regions in stratified fluids [4], disturbances induced by moving objects [5–7], flow past topography [8], including Lee waves locked on to the topography [9] and due to perturbations induced by contiguous turbulent regions [10]. Laboratory studies of internal waves abound [11], and more recently increased attention has been given to internal waves generated by stationary and moving bodies in stratified fluids, with applications to oceanic submersibles and underwater facilities [12]. In this class of problems, uniformly stratified flow of horizontal velocity U and buoyancy frequency N past a body of characteristic dimension D is considered (equivalent to a body moving with the same speed in opposite direction in a quiescent stratified fluid). The major dimensionless parameters governing this problem are the Froude Number Fr and the Reynolds number Re, defined as U Ud g d Fr = and Re = , with N = ND (z) dz where is the density, g the gravitational acceleration, and axis Oz is vertically downward. A host of phenomena appears for varying Fr and Re, regime diagrams for which have been plotted by Boyer et al. [5] for a 2D cylinder moving horizontally in a stratified fluid, Lin et al. [6, 13] for spheres moving horizontally, and Hanazaki et al. [14] for spheres moving vertically. When Fr is large, relatively long internal waves are generated in the lee of the obstacle, whereas for smaller Fr upstream propagating wave modes are possible. The latter takes the form of columnar modes where the wave number is vertical and fluid motions are horizontal. As such, motions do not feel the density stratification and fluid parcels propagate horizontally as intrusions with no oscillations [4, 15]. Boyer et al. [5] showed that the effect of the obstacle is felt upstream for a distance that is dependent on Fr and Re, and in the inviscid limit the effect of the obstacle is felt in the full domain. Although stratified flow over bluff bodies have been mainly conducted using spheres and cylinders, other canonical geometries have also been used in the geophysical context. These include flow over half sphere [16], flat plates [17], and thin vertically barriers [18]. In most of these studies, the steady motion of the body/flow has been considered. The corresponding studies on impulsively started case are meager. It is this latter problem that is of interest here, where impulsively started flow past a thin vertical strip is considered. The general structure of such bluff-body flows, including upstream influence, internal waves, and downstream wake has been studied both theoretically [15, 19] and experimentally [9, 20–22]. At moderate Froude and Reynolds numbers, the formation of wake vortices and vortex arrays have been observed, in addition to wave propagation. Castro [23] studied weakly stratified flow past a flat plate located normal to the flow in a range of small Reynolds and Froude numbers numerically, wherein the gravity waves are essentially suppressed. Several downstream vortex regimes were identified, and the results were compared with various flow regimes predicted by theoretical analysis. A general agreement was noted in many cases, at least qualitatively, and existence of some multiple (i.e. non-unique) solutions was demonstrated. Sutherland and Linden [18] conducted experiments in a recirculating shear flow water tunnel by placing a thin tall vertical barrier in a non-uniformly stratified flow, and investigated how coherent vortex structures emanating at the edges can excite internal waves. In agreement with numerical simulations, large amplitude internal waves were found to form when the wake mixing region is weakly stratified compared to ambient stratification. Flow visualization and quantitative measurements with a ‘Synthetic Schlieren’ and ensuing wavelet transforms analysis showed that the wave packets radiating out of the mixing region has a narrow frequency range. These wave packets carry a measurable momentum flux into the ambient flow, which in turn has a positive feedback on wave radiation out of the wake region at this particular frequency. The purpose of this paper is to calculate stratified flow past bluff bodies, in particular an impulsively started thin vertical barrier moving at a constant horizontal speed, using an existing Copyright 䉷 2011 John Wiley & Sons, Ltd.

Int. J. Numer. Meth. Fluids 2012; 68:451–466 DOI: 10.1002/fld

NUMERICAL MODELING BY UNIFORM STRATIFIED FLOW

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numerical code to obtain high space–time resolution numerical data sets that can be validated by experimental work. Once validated, the code can be used for other geometries and types of flows that may not be readily amenable to laboratory and field experimentations. Only few comparisons of stratified flow past vertical barriers with experimental data have been reported so far, and this paper attempts to do so albeit in a limited parameter range. Given the lack of past experimental data on starting stratified flows, further experiments were conducted as a part of this study using a vertical barrier moving in a stratified fluid. A prognostic numerical model was first set up based on primitive equations in fluid mechanics, which was solved using a CFD code. The model evaluation was focused on the properties of internal waves, and further comparisons are planned as high resolution experimental data become available on velocity and density fields in the future. In Section 2, the governing equations and dimensionless variables are introduced for the steady flow problem. The numerical technique is described in Section 3, followed by laboratory experimental details in Section 4. The numerical results are given in Section 5 and their comparison with experimental results in Section 6. The concluding remarks are given in Section 7.

2. GOVERNING EQUATIONS AND SCALING We consider disturbances induced by an impulsively started thin vertical strip of height h with velocity Ub in an incompressible, isothermal, viscous, uniformly stratified fluid layer of depth L z and horizontal length L x (Figure 1). The Cartesian coordinate frame is placed at the top of the layer with the axis Oz directed downward. The undisturbed density profile is exponential 0 = 00 exp[z/], characterized by the stratification length scale = |d ln 0 /dz|−1 , where 00 = 3 0 (0) = 1008.9 kg/m is a reference density. The buoyancy (Brunt–Väisälä) frequency is thus N = 2/Tb = g/, where Tb is the buoyancy period. Owing to the symmetry, the flow is considered two-dimensional (y independent) and the Reynolds numbers of most computations are small enough to ensure laminar motion. As the Reynolds number increases, beyond Reynolds numbers of about 200 the inertial effects become important (weakly turbulent), which are accounted for by invoking Smagorinsky closure (see Table IV). It should be noted that some important three-dimensional effects, such as spanwise instabilities and structures, are ignored here, studies of which require three-dimensional numerical simulations. Such intricacies will be pursued in the future studies (for example, see Dauchy et al. [24] for the non-stratified case without lateral confinement). The 2D averaged governing equations to the Boussinesq approximation are *u¯ j =0 *x j *u¯ i *u¯ i 1 * p¯ * − + gi3 +2 [(+t )¯ij ] = −u¯ j *t *x j 0 *xi 0 *x j 1 *u¯ i *u¯ j ¯ ij = + 2 *x j *xi 0

x

z

Lz

h

Ub

ρ ( z ) = ρ exp(z / Λ )

Lx

Figure 1. A schematic of the flow configuration and boundary conditions. Copyright 䉷 2011 John Wiley & Sons, Ltd.


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* * S¯ * S¯ * S¯ (+ ) + = −u¯ j *t *x j *x j *x j

(1)

= 0 [1+s ( S¯ − S¯0 )] t = (Cs )2 |¯| || = |2(ij ij )|1/2 ij = 1

if i = j,

ij = 0

if i = j,

where u¯ i is the component of filtered velocity in i (= 1, 2, 3) direction, p the pressure, S the salinity, s the salt contraction coefficient, and t are the kinematic molecular and turbulent viscosity coefficients, and and t the molecular and turbulent salt diffusion coefficients, respectively. For Smagorinsky-type turbulent closure used in large eddy simulations, Cs = 0.2 and = 2.7×10−3 m are chosen, and the turbulent Schmidt number is fixed to 1. The boundary conditions on the channel end walls are *w = 0, *x

u = 0,

*S * = = 0 at x = 0, L x *x *x

(2)

which allow selected disturbances to leak out from the end walls. On the other hand, the boundary conditions at the top and bottom of the channel are *u = 0, *z

w = 0,

*s * = =0 *z *z

at z = 0, L z

(3)

where free surface conditions are assumed. The plate is impulsively started, where at t>0 the velocity of its surface has an x-direction velocity of Ub . On dimensional grounds, the governing parameters for the 2D Boussinesq problem are Ub , h, , L x , L z , N0 , , , x, z, where N0 is the buoyancy frequency at the plate mid-depth (z = L z /2) and thus the relevant dimensionless parameters are Ub Ub h Ub h Lx Lz x z Fr = , Re = , Pe = , , , , , (4) N0 h h h h h h where Pe is the Peclet number. With the assumption of boundaries located far from the body during the period of interest, the disturbances are assumed to be not influenced by the boundaries. The following dimensional parameters determine the problem Fr,

Re,

Pe,

, h

x , h

z . h

(5)

Note that if the Boussinesq approximation is not called in, then the density variations directly come into play and the important parameters are Ub , h, , L x , L z , g, 0 , , , , x, z, where is a density scale, say the difference between the upper and lower boundaries of the channel. Thus a possible set of dimensionless parameters are ⎛ ⎞ Ub ⎠ Ub h Lx Lz x z Ub h ⎝ Fr =

, Pe = , , , , (6) , Re = , , h h h h h 0 gh 0

which can be simplified to obtain (5), using the aforementioned arguments.

3. COMPUTATIONS The numerical code used herein was adapted by Berrabaa et al. [25] from the JETLES solver (courtesy of Verzicco and Orlandi [26]) developed for LES and direct numerical simulation of Copyright 䉷 2011 John Wiley & Sons, Ltd.



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variable density flows. The second order finite difference discretization in combination with a third order Runge-Kutta time marching procedure used in the code was quite efficient for fast unsteady flows at a reasonable computational cost. The code (which is 3D) has been rewritten in Cartesian mesh and equation of salinity transport has been added to account for density effects [25]. The mean flow is assumed two dimensional, as shown in Figure 1. The code, however, was run on a quasi 3D mode with 9 grid points along the cross direction. A vertical strip of height h, started impulsively at t = 0, is moving from left to right at a fixed velocity Ub in a fluid with the specified density profile 0 (z). The computational domain is L x × L z . The horizontally moving grid was modeled using Dirichlet boundary conditions imposed on the grid cells that define the horizontal and vertical geometry of the moving strip. The boundary conditions at the inlet and outlet boundaries are non-reflective as given in (2) and the upper and lower boundaries of the computational domain have free reflective boundaries (3). Computations of impulsively started flow were limited to a few seconds of initial evolution, as in laboratory experiments, to focus on internal wave radiation without undue influence of boundaries. Although the Reynolds and Peclet numbers were kept small to moderate, when operating at the higher end of the Reynolds number (Re ∼ 150), non-linear effects such as vortex shedding and turbulence are expected. To account for this perceived weakly turbulent flow, as mentioned, the generic Smagorinsky turbulence closure model was applied. The size of the numerical grid was selected after several numerical tests to enable resolving the wave field, rather than near-wake features, which has been the foci of previous work (e.g. [23, 27]). As such, near wake fine-scale structure, which has been reported in previous experiments as well as present laboratory experiments, were not considered in the analysis or in discussions of the rest of the paper. Moreover, the formation of certain wake features, including thin layers surrounding the body and in wakes, occurs over time scales longer than those for the waves to reach boundaries [1], and hence the study of such flow phenomena (though rich in physics) was untenable. The computational domain was discretized using finite differences to form a 514×602 Cartesian grid with a mesh size of 4.3 mm in the horizontal and 1 mm in the vertical, and the time step was 0.5×10−2 s. The computational time was about 30 days on a 4 CPU Intel Pentium 3.2 GHz HP Compaq dx 6120 MT machine.

4. EXPERIMENTAL FACILITY The experiments were conducted in a rectangular transparent tank (220×40×60 cm3 ) filled with linearly stratified brine. Fixed to the side walls of the tank were optical windows for flow illumination, visualization, and observations; a Russian-made Schlieren instrument IAB-458 with a 23 cm diameter field of view was used. As in conventional Schlieren, this employs various optical diaphragms, a vertical thin illuminating slit and a vertical flat knife or a visualizing thread. The slit/knife method is more sensitive and able to produces discernible and contrasting images of flow disturbances of varying strengths [28]. The method helps to distinguish lines of internal wave crests (dark) and troughs (gray), and produces images of interfaces on a background with strong internal waves at a spatial resolution better than 0.1 mm. In a number of experiments, neutral density markers were used to study density and velocity disturbances. The marker was a free falling sugar crystal or small rising gas bubbles. The estimated accuracy of velocity measurements is better than 5% with a spatial resolution of about 0.25 mm (thickness of the marker). In quiescent environments, after their introduction, the density markers could be used for times up to 200 s. Prior to each experiment, the buoyancy period Tb was measured using an in-house made conductivity probe placed near selected density markers with an accuracy ∼ 5% [22]. The probe was calibrated using ‘sweep-oscillations’ or ‘up-down’ movements over around 1 cm. The inferred density profile was found to be roughly linear, but it could be better approximated to an exponential theoretical profile with a suitable . It is necessary to underscore that both the density and velocity fields were ‘visualized’ over the entire water depth in a longitudinal plane without introducing solid particles that may change the physical properties of the fluid media. An approximately Copyright 䉷 2011 John Wiley & Sons, Ltd.


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two-dimensional vertical strip of height h = 2.5 cm was then towed along the tank by a carriage installed on rails above the tank, and flow in the central part of the observing window was recorded and internal wave parameters and deformation of density profiles were inferred.

5. ANALYSIS OF NUMERICAL RESULTS When the body was impulsively started, the initial perturbation generated near the body propagates as shown in the sequence of isotachs of Figure 2, where the temporal evolution of the vertical velocity is illustrated at three dimensionless times t/Tb = 1.32, 3.96, and 6.6. The observed flow patterns are typical of transient internal waves generated at an impulsively started body in a stably stratified fluid [25]. In each quadrant of the x–z plane surrounding the body, the number of loops reflects the number of half periods (e.g. time between the crests and troughs; for example, equal to 3 for t/Tb = 1.32 in Figure 2(a). As a result of the unidirectional movement of the barrier and non-linear density distribution, the phase patterns around the objects are asymmetric with respect to the vertical barrier as well as with respect to the horizontal canter plane of the object. In Figure 2(a), the wave propagation downstream (to the left of the figure) is concentrated in two directions corresponding to the group velocity of propagating transient waves generated by the body. For short internal waves where the wavelength is less than the depth of the tank, the dispersion relation is [11, 29] N |k x | =

= N sin

k x2 +k z2

w / Ub

w / Ub

-8

0.8

-6

0.6 -4

0.4

1

-6

0.8

-4

0.6 0.4

0

0

2

-0.2 -0.4

4

-8

-2

0.2 z/h

z/h

-2

(7)

0.2

0

0

2

-0.2 -0.4

4

-0.6

-0.6 6 8 -10

-0.8 -5

01.716 5

15

10

(a)

20

25

30

6

-0.8

8 -10

-1 -5

0

5.148 10

(b)

x/h

15

20

25

30

x/h w / Ub

-8 -6

1 1 0.8

-4

0.6 0.4

z/h

-2

0.2

0

0

2

-0.2 -0.4

4

-0.6

6 8 -10

-0.8 -1 -5

0

(c)

5 8.58

15

20

25

30

x/h

Figure 2. Isotachs of vertical velocity h = 2.5 cm, Tb = 12.5 s, U = 0.26 cm/s, Re = 65, Fr = 0.21, C = 1550: (a) t/Tb = 1.32; (b) t/Tb = 3.96; and (c) t/Tb = 6.6. Copyright 䉷 2011 John Wiley & Sons, Ltd.


457


Table I. (Figures 2(a)–(c)): Location and value of maximum and minimum vertical velocity.

wmax /Ub (xb − xmax )/ h z max / h wmin /Ub (xb − xmin )/ h Z min / h

t/Tb = 1.32

t/Tb = 3.96

t/Tb = 6.6

1.0083 0.5 −0.5 −0.98633 0.5 0.5

1.1027 0.5 −0.5 −1.0754 0.5 0.5

1.1204815 0.5 −0.5 −1.0991141 0.5 0.5

wmax , maximum value of vertical velocity; wmin , minimum value of vertical velocity; xmax , horizontal position of maximum vertical velocity; xmin , horizontal position of minimum vertical velocity; xb , horizontal position of the strip; z max , vertical position of maximum vertical velocity; Z min , vertical position of minimum vertical velocity.

Table II. (Figures 2(a)–(c)): Angular position of wave direction (angle is measured from the horizontal). (Figure 2(a)) t/Tb = 1.32

1

2

16◦ (Figure 2(b)) t/Tb = 3.96

1

2

6◦

5◦

3

16◦

(Figure 2(c)) t/Tb = 6.6

1

3

44◦

2

15◦

3

25◦

4

26◦

4

32◦

66◦

5

36◦

5

39◦

6

45◦

6

46◦

7

50◦

8

57◦

7

55◦

9

61◦

8

64◦

10

66◦

11

70◦

74◦

12

75◦

13

80◦

where is the slope of the wave vector k = (k x , k x ) to the vertical. Then the group Cg and phase cph velocities are defined as N k z2 signk x * (k x , k z ) N k z |k x | N sin

* (k x , k z ) (8) , |Cg | = = , , Cg = 2 2 3/2 2 2 3/2 |k| (k x +k z ) (k x +k z ) *k x *k z cph =

k . k2

(9)

The positions of the maximum and minimum vertical velocities are given Table I for the three non-dimensional times and Table II gives the angular distribution of initial transient waves (as evident from the lobes). Note that transient internal waves propagate ahead of the moving barrier. The maximum particle velocities of the waves in the upstream are found in the vicinity of the edge of the strip. The interferences of transient waves gradually start to manifest in the wake region, which is the region swept by the body from its initial to instantaneous position along the trajectory. As a result, iso-phase (or phase) lines of transient waves gradually transform from circular to semi-circular shapes centered at the instantaneous position of the obstacle, taking the form of lee waves locked on to the body; the projection of phase velocity on the trajectory of the strip is the same as the towing velocity. These lee waves are coupled to transient waves propagating ahead of the strip. With time, the number of energetic waves increases, and at dimensionless time = t/Tb = 3.96 (Figure 2(b)) about seven half waves are observed along the trajectory delimited by (vertical) dashed lines. They occupy the entire interference domain bounded by a semi-circular contact surface between the outer propagating transient waves and lee waves and the body. On the left side, starting from x = 0, there are a set of transient waves, the energy of which is propagating along straight lines. As a result, a quieter region is formed downstream of the source, and because of the differential speeds of the waves with different inclinations, motions in this region tend to be anisotropic. Copyright 䉷 2011 John Wiley & Sons, Ltd.


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u / Ub -8 0.8

-6

0.6 0.4

-4

0.2

z/h

-2

0

0

-0.2 -0.4

2

-0.6

4

-0.8 -1

6

-1.2 8 -10

-5

0

5 8.58

15

20

25

30

x/h

Figure 3. Isotachs of horizontal velocity h = 2.5 cm, Tb = 12.5 s, U = 0.26 cm/s, Re = 65, Fr = 0.21, C = 1550, t/Tb = 6.6.

At = t/Tb = 6.6, 13 loops of stationary lee waves fill the domain between the starting and the new position of the strip (Figure 2(c)). In the far wake of the left side of the initial position (x = 0), no reflection of transient waves is observed due to efficiently absorbing boundary conditions. Also note the development of a quiescent region because of the wave radiation. As is evident from Figures 2(c) and 3, at large times the phase lines in front of the object tend to be horizontal, and hence from (7) the wave numbers tend to be vertical ( = 0). This implies the formation of approximately zero-frequency internal waves [4]. A blocked region is formed in front of the object that is moving with the same speed as the body in the near field upstream, and this is owed to the fact that the kinetic energy of the motions field is insufficient to overcome stable stratification, rise/dip, and pass over/under the strip. However, due to viscous diffusion, the shear layer between the blocked region and the surroundings become weak after some distance and the blocked region is ‘dissolved’ further upstream. Boyer et al. [5] proposed that the length of the upstream propagating fluid slug is given by / h ≈ CReFr−2 , where C ∼ 10−2 . Substituting the values Fr = 0.21 and Re = 65 for Figure 3, it is possible to obtain / h ≈ 15, which is close to what is observed in Figure 3. Also note that, as is evident from Figure 2, lee wave activity behind the body increases with time, which acts as a source of propagating waves to the exterior region. With time, the lee wave activity adjusts so that the phase in the anterior of the body is horizontal and an upstream traveling slug of fluid is generated (Figures 2(a)–(c)). As mentioned, the coupling between the propagating and lee waves is dynamically important, and as the number of lee waves changes the properties (wave length and frequency) of propagating waves also change. The angles of first upstream transient wave isoclines to the horizontal (e.g.

1 in Table II) is then decreasing to form the upstream influence, whereas in the wake close to the horizontal body centerline the phase lines are vertical, thus trapping the energy (small group velocity oscillations occur at the buoyancy frequency with a minimum wavelength =Ub Tb ). As the distance from the horizontal centerline increases, wave isoclines bend, allowing waves to propagate at an angle and take energy away from the region. The horizontal component of velocity shown in Figure 3 is different from the vertical component shown in Figure 2, wherein the vertical component is anti-symmetric with respect to the horizontal axis whereas the horizontal component is symmetric. Wave surfaces are more pronounced in Figure 3, including the completely blocked area attached to the strip that forms a wedge domain and a complex downstream wake bounded by lee waves. Above and below the downstream wake, there are two strips with very low horizontal velocity. The periodic wave structures are superposed on uniform mean motion in the near wake. The far upstream disturbance is separated into two branches indicating the rays of transient internal waves. The near field completely blocked fluid is Copyright 䉷 2011 John Wiley & Sons, Ltd.



-8

459

6

-6

5

-4

4 3

z/h

-2

2

0

1

2

0 -1

4

-2

6 8 -10

-3 -4 -5

0

5

8.58

15

20

25

30

x/h

Figure 4. Iso-gradient lines of horizontal velocity h = 2.5 cm, Tb = 12.5 s, U = 0.26 cm/s, Re = 65, Fr = 0.21, C = 1550, t/Tb = 6.6.

formed by superposition of two rays. A momentum transfer zone between the lee waves and initial transient motions is also observed. The wave pattern is more pronounced above the strip than below because of the non-linear stratification. An iso-gradient graph of the horizontal velocity component is shown in Figure 4. In contrast to its velocity counterpart in Figure 3, which uniformly decays with the distance from the horizontal centerline, the velocity gradients achieve the maximum values in an expanding circular domain in contact with the spatial transient zone between lee and transient waves. Another region of pronounced velocity gradients is observed surrounding the horizontal centerline, wherein the frequency of oscillations is the buoyancy frequency, the group velocity of waves is null and trapped energy of the oscillations is decreasing due to viscosity. Another physical parameter of interest is the density, the spatial field of which is shown in Figure 5. The transient waves on the left side (downstream) are more visible, and the wave field as represented by the isopycnals is anti-symmetric with respect to the horizontal centerline due to the non-linearity of stratification and the shape and position of the obstacle. The blocking effect in the anterior of the body manifests a weakening of the density gradient due to spatial adjustment of isolines, which in part is due to diffusion-induced mixing near the edges of the upstream slug. The convergence of this partially mixed region in the aft of the obstacle creates a jet-like flow near the center of the wakes, as is evident from Figure 3 and in laboratory experiments to be described later. The location of the maximum density variations correlates with the velocity gradient in Figure 4 than with the vertical and horizontal velocity components in Figures 2 and 3, respectively.

6. COMPARISON BETWEEN EXPERIMENTS AND COMPUTATIONS A Schlieren image of flow generated by towing a vertical strip in a stratified tank, as discussed in Section 4, is presented in Figure 6(a). To produce the image, an illuminated slit was placed vertically, and a Foucault knife placed in a vertical position (as a light cutting diaphragm) produced the Schlieren effect [22]. In this technique, the illumination of the flow image is proportional to the horizontal component of the index of refraction, which is linearly dependent on the density gradient [28]. The photos so obtained (Figure 6(a)) allowed qualitative comparisons of density gradients with computational results that are shown in Figure 6(b). The boundaries between black and white strips correspond to crests and troughs of internal waves. Only four lee waves and their spatial transition to zero-frequency waves that are responsible for generating upstream influence are visible in Figure 6(a). The lee waves are evident as confined to a semicircular area, spatial transition between lee and transient waves is well captured and the Copyright 䉷 2011 John Wiley & Sons, Ltd.


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ρ ρ

00

-4

1.014 -3 1.012 -2 1.01

z/h

-1

1.008

0 1

1.006

2

1.004

3

1.002

4 -2

0

2

4

6

8 8.58 10

12

x/h

Figure 5. Isopycnals around the body h = 2.5 cm, Tb = 12.5 s, U = 0.26 cm/s, Re = 65, Fr = 0.21, C = 1550, t/Tb = 6.6.

phase lines of transient waves to the left of starting point (x = 0) appear as straight lines. The wave length is in satisfactory agreement with the prediction of the linear theory =Ub Tb = 1.74 cm. In agreement with the computational results (Figure 6(b)), the wave phase surfaces upstream and downstream of the body are matched in the experiments. In the experimental downstream wake, it is possible to observe a set of cusps (a herring-bone pattern) skirting away from the direction of motion, which could not be resolved in the computations. On the other hand, transient waves visible in computations are not evident in the experiments due to the limitation of the extent of the high-resolution optical window. In all, while there is a general agreement between the observations and computations, each visualization technique also exhibits its own specific features, thus complementing information provided by the other. An advantage of computations is that they provide variables at every grid point in the flow volume, but suffer the drawback of needing a turbulence model and idealized boundary conditions. In Figure 7, a more quantitative comparison is made between computations and observations, where vertical profiles of horizontal velocity at different upstream locations are indicated based on the distortion of marker lines. In the background, a Schlieren image is produced using a vertical illuminated slit and a vertical filament as the cutting diaphragm [28]. In contrast to the conventional Schlieren that employs a knife cutting diaphragm (Figure 6(a)), where only large-scale waves can be observed as wide curved surfaces, slit-filament method can demonstrate fine structure disturbances ahead and past the obstacle. In the images (Figure 7(a)), the crests are curved dark lines and the troughs are double gray lines that end at the thin (jet-type) wake of the moving body as evident in Figure 7(a). Horizontal velocity profiles reconstructed using sugar-marker displacements are shown in Figure 7(b), where the horizontal velocity in all profiles is normalized by the obstacle velocity Ub . The horizontal bar shows the unit magnitude, which can be used to gage the relaxation of upstream-blocked velocity profiles, and as discussed in Boyer et al. [5] the relaxation is manifested by viscous diffusion. Spatial oscillations of velocity profiles on either side of the upstream slug are indicative of near zero-frequency internal waves that show fluid layers sliding on each other in opposite directions [4]. Sloping solid lines indicate the location of the minimum and maximum velocities ahead of the obstacle corresponding to the wave rays in Figures 2 and 3. The wave velocity gradually decreases with the distance from the source. Computed vertical profiles of horizontal velocity are given in Figure 7(c), which are similar to those observed in Figure 7(a). The dashed lines connecting the points of maximum (black), minimum (red), and zero velocity (in blue) are crossing the horizontal centerline at three different points, and this is a result of vertical diffusion of horizontal momentum through the shear layer. Copyright 䉷 2011 John Wiley & Sons, Ltd.



461

(a) h ∂ρ ρ00 ∂ x -8

x 10-4 5

-6

4 3

-4

2 z/h

-2

1 0

0

-1

2

-2

4

-3 -4

6

-5 8 -10

-5

0

(b)

5

8.58

15

20

25

30

x/h

Figure 6. (a) A Schlieren image of stratified flow past a vertical strip (h = 2.5 cm, Tb = 17.4 s, U = 0.1cm/s). (b) An iso-density gradient map for h = 2.5 cm, T b = 12.5 s, U = 0.26 cm/s, Re = 65, Fr = 0.21, C = 1550, t/Tb = 6.6.

In theory [28], experiments [22], and current numerical modeling, initial phase lines correspond to rays of transient internal waves set up by the body movement. The horizontal distance between the points of intersections of phase lines with the centerline is an indicative of the length of the imaginary source of wave generation. In physical experiments the length between the lines of maximum and minimum velocity L is 6.60 cm, whereas in numerics the value is 8 cm. For experiments where the strip √ height is 2.5 cm, the length of lee waves L is 3.25 cm, the internal viscous length scale L = 3g/N is 4.28 cm, and the length of the blocked zone from computations L b is 2.0 cm. Some of the computed and experimental ratios are given in Table III. From Table III it follows that the ratio L/L is the minimum, so it appears that viscosity as well as buoyancy play important roles in the formation of the blocked zone and the general structure of the flow pattern. Table III shows a comparison between experiments and numerics of the length of the blocked region normalized by various length scales. The two are closest when normalized by the internal viscous length scale L , supporting the idea that viscous and buoyancy effects play an important role in the upstream slug formation. As shown in Figure 8, the normalized length of the upstream blocked region is proportional to inverse Froude number for Fr>0.2, which is not consistent with Boyer et al. [5] result as / h ≈ CReFr−2 , perhaps because the latter is valid only under low Fr conditions. Both the numerical and experimental results are in agreement with Fr−1 scaling. The velocity profiles characterizing the upstream influence expand outward (i.e. velocity amplitude Copyright 䉷 2011 John Wiley & Sons, Ltd.


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(a) 10 1

2

3

4

5

6

y, cm

5

0

-5 u/U 1

-10 0

5

(b)

10

15

x/ h -4

1 u/Ub

-3 -2

z/h

-1 0 1 2 3 4 -13.4 -10.2

(c)

0

2.7 5.1

9.2

13.1

17.5

x/h

Figure 7. Flow patterns and velocity profiles ahead of a strip moving in a stratified fluid as obtained in the laboratory by a Schlieren and also computationally (h = 2.5 cm; Tb = 12.5 s; U = 0.26cm/s; Fr = 0.21): (a) Schlieren image of the flow; (b) distortion of vertical markers, illustrating the vertical profile of horizontal velocity extracted from the picture in (a); (c) computed horizontal velocity profiles for upstream locations −h = 2.5 cm, Tb = 12.5 s, U = 0.26cm/s, Re = 65, Fr = 0.21, C = 1550, x/ h = 2.7−5.1−9.2−13.1−17.5.

decays) with the upstream distance, which can be attributed to viscous diffusion. In addition, strong density gradient regions also thicken upstream due to salt diffusion (Figure 7(a)). The vertical displacement of fluid parcels, as measured by a conductivity probe placed at different distances from the centerline, is shown in Figure 9 in a coordinate frame fixed to the obstacle. Copyright 䉷 2011 John Wiley & Sons, Ltd.



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Table III. (Figures 7(b) and (c)): Ratio of different scales. Experimental results

Numerical results

2.64 2.03 1.54 3.3

3.2 2.46 1.87 4

L/ h L/L L/L L/L b

1.2

Lb / h

0.8

0.4

0 0

5

10

1/ Fr

Figure 8. The length of the completely blocked fluid area versus inverse Froude number ahead of the strip (h = 2.5 cm; Tb = 12.5 s): 䊉 Numerical results and Experimental result.

Δ,cm

6

1

z, cm

4

2

0

0

10

20

30

40

50

x, cm

Figure 9. Measured vertical displacements of fluid particles past vertical strip (Tb = 14 s; h = 2.5 cm; U = 0.27 cm/s; Fr = 0.24).

The position of the strip (x ∼ 29 cm) and the scale of displacement (upper left) are also shown. The loci of maximum and minimum displacements are consistent with the crests and troughs of the waves shown in Figures 6(a) and 7(a) based on experiments and in Figures 2 and 3 from computations. The number of crests and troughs visible on Schlieren and computational images are larger than what is evident from Figure 9 due to the low sensitivity of the probes compared to Schlieren technique and numerical simulation. For example, the computed variation of the normalized density (/00 ) at the horizontal level h = 1 cm, which is inaccessible for measurements due to interference with the body, is shown in Figure 10. The resolved features in Figure 10 Copyright 䉷 2011 John Wiley & Sons, Ltd.


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1.039 1.038

ρ ρ00

1.037 1.036 1.035 1.034 -40

-30

-20

-10

0

10

20

30

x/h

Figure 10. Computed variations of normalized density at the horizontal level h = 1 cm. Table IV. Parameter values describing a sequence of five simulations. L X (m) 2.2 2.2 2.2 2.2 2.2

L Z (m) 0.6 0.6 0.6 0.6 0.6

NX 514 514 514 514 514

NZ

(m2 /s)

Ub (m/s)

h (m)

Tb (s)

Re

Fr

602 602 602 602 602

10−6

0.26×10−2

0.025 0.025 0.025 0.025 0.010

12.5 12.5 12.5 12.5 3

65 75 102.5 225 200

0.21 0.24 0.33 0.72 0.95

10−6 10−6 10−6 10−6

0.30×10−2 0.41×10−2 0.90×10−2 2×10−2

are consistent with the general spatial trends of experimental isopycnals shown in Figure 9. The displacement of isopycnals away from the obstacle is due to disturbances propagating away from the obstacle, and hence is determined locally. As a result, after the passage of the obstacle, such displacements settle at a frequency determined by the local buoyancy frequency and initial disturbance. At large times, these waves decay owing to viscous dissipation (Table IV).

7. CONCLUSIONS The numerical code for inhomogeneous turbulent flow calculations developed by Berrabaa et al. [25], based on the OLES code proposed by Verzicco and Orlandi [26], was used to study flow past obstacles in stratified fluids. The aim was to validate the code using laboratory experiments so that it can be used to compute a suite of kindred flows. As a first step, a canonical flow configuration consisting of a vertical strip towed in a stratified fluid at relatively low Reynolds numbers was used, where the density in the fluid is exponentially varying (but close to a linearly stratified fluid). Fields of velocity, density and their gradients were computed and visualized and compared with experimental data. In addition, the computations provided finer details of the flow that could not be evinced experimentally, thus augmenting the experimental results. In general, a good agreement was found between computations and experiments, both quantitatively and qualitatively. The performance of the code was found to be acceptable in predicting fast unsteady wave motions and in simulating important physical processes of stratified flow over topography. Further refining of computing mesh is expected to help resolve additional small-scale flow features produced by viscous and diffusivity effects that are clearly evident from experiments but could not be resolved in simulations. In addition to validating the code, the present work provided useful physical insights into internal wave generation in a towed body system. At the Froude numbers used, a lee wave field was generated by the fluid parcels that rise and fall at the obstacle edge, and a slug of fluid was pushed Copyright 䉷 2011 John Wiley & Sons, Ltd.



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ahead of the body due to upstream influence. As pointed out by Mannis [30] and DeSilva and Fernando [4], the generation of an upstream slug of fluid is associated with the generation of (near) zero-frequency internal waves. Surrounding the body, these two types of waves were found to be connected by iso-phase (or phase) lines through a transition zone wherein the phase lines of lee waves bend forward to produce those of (near) zero-frequency waves. The latter constitute sliding motions between layers moving in opposite directions. The influence of ensuing shear, propagating waves and perhaps viscous effects cause phase lines away from the body to be inclined. In addition, some of the isopycnals flowing over/under the obstacle meet in the wake to produce a rich set of phenomena, as discussed by Boyer et al. [5]. Propagating waves are generated at the shear layers in the outer regions of the body wake, which carry energy away from the wake region to cause gradual decay of vertical motions in intermediate and far wake regions. ACKNOWLEDGEMENTS

The paper was partly financially supported by the RFBR (grant 08-05-00473 and 08-05-90434). PACA region grant for International Research in Mediterranean area. HJSF was partly supported by the ONR (turbulence and wakes program). REFERENCES 1. Staquet C, Sommeria J. Internal gravity waves: from instabilities to turbulence. Annual Review of Fluid Mechanics 2002; 34:559–593. 2. Fernando HJS. The evolution of a turbulent patch in a stratified shear flow. Physics of Fluids 2003; 15(10): 3164–3169. 3. Phillips OM. Dynamics of the Upper Ocean. Cambridge University Press: Cambridge, 1977. 4. DeSilva IPD, Fernando HJS. Experiments on collapsing patches in stratified fluids. Journal of Fluid Mechanics 1998; 358:29–60. 5. Boyer DL, Davies PA, Fernando HJS, Zhang X. Linearly stratified flow past a horizontal circular cylinder. Philosophical Transactions of the Royal Society A 1989; A328:501–528. 6. Lin Q, Lindberg WR, Boyer DL, Fernando HJS. Stratified flow past a sphere. Journal of Fluid Mechanics 1992; 240:315–354. 7. Chashechkin YD, Gumennik EV, Sysoeva EY. Transformation of a density field by a three-dimensional body moving in a continuously stratified fluid. Journal of Applied Mechanics and Technical Physics 1995; 36(1):19–29. 8. Baines PG. Topographic Effects in Stratified Flows. Cambridge University press: Cambridge, 1997. 9. Long RR. Some aspects of the flow of stratified fluids: I. A theoretical investigation. Tellus 1953; 5:42–58. 10. Fernando HJS, Hunt JCR. Turbulence, waves and mixing at shear free density interfaces: Part 1. Theoretical model. Journal of Fluid Mechanics 1997; 347:197–234. 11. Turner JS. Buoyancy Effects in Fluids. Cambridge University Press: New York, 1977. 12. Meunier P, Spedding GR. Stratified propelled wakes. Journal of Fluid Mechanics 2006; 552:229–256. 13. Lin Q, Boyer DL, Fernando HJS. The vortex shedding of a streamwise-oscillating sphere translating through a linearly stratified fluid. Physics of Fluids 1994; 6(1):239–252. 14. Hanazaki H, Kashimoto K, Okamura T. Jets generated by a sphere moving vertically in a stratified fluid. Journal of Fluid Mechanics 2009; 638:173–197. 15. Janowitz GS. The slow transverse motion of a flat plate through a non-diffusive stratified fluid. Journal of Fluid Mechanics 1971; 47(1):171–181. 16. Brighton PWM. Strongly stratified flow past three-dimensional obstcles. Quarterly Journal of the Royal Meteorological Society 1978; 104:289–307. 17. Brown SN. Slow viscous flow of a stratified fluid past a finite flat plate. Proceedings of the Royal Society A (306) 1968; 1485:239–256. 18. Sutherland BR, Linden PF. Internal wave excitation from stratified flow over a thin barrier. Journal of Fluid Mechanics 1998; 377:223–252. 19. Trustrum K. An Oseen model of the two-dimensional flow of a stratified fluid over an obstacle. Journal of Fluid Mechanics 1971; 50(1):177–188. 20. Miles JW. Lee waves in a stratified flow Part 1. Thin barrier. Journal of Fluid Mechanics 1968; 32:549–567. 21. Davis RE. The two-dimensional flow of a stratified fluid over an obstacle. Journal of Fluid Mechanics 1969; 36(1):127–143. 22. Chashechkin YD, Mitkin VV. Experimental study of a fine structure of 2D wakes and mixing past an obstacle in a continuously stratified fluid. Dynamics of Atmosphere and Oceans 2001; 34:165–187. 23. Castro IP. Weakly stratified laminar flow past normal flat plates. Journal of Fluid Mechanics 2002; 454:21–46. 24. Dauchy C, Dusek J, Fraunié P. Primary and secondary instability in the wake of a cylinder with free ends. Journal of Fluid Mechanics 1997; 332:295–339. Copyright 䉷 2011 John Wiley & Sons, Ltd.


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25. Berrabaa S, Fraunié P, Crochet M. 2D large eddy simulation of highly startified flow: the spetwise structure effect. Advances in Computation: Theory and Practice, vol. 7. Nova Science Publishers: New York, 2001; 179–186. 26. Verzicco R, Orlandi P. A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. Journal of Computational Physics 1996; 123(2):402–414. 27. Carte G, Dusek J, Fraunié P. Numerical simulation of the mechanism governing the onset of the Bénard–von Karman instability. International Journal for Numerical Methods in Fluids 1996; 23:735–785. 28. Chashechkin YD. Schlieren visualization of a stratified flow around a cylinder. Journal of Visualization 1999; 1(4):345–354. 29. Lighthill J. Waves in Fluids. Cambridge University Press: Cambridge, 1978. 30. Mannis PC. Intrusion into a stratified fluid. Journal of Fluid Mechanics 1976; 74(3):547–560.

Copyright 䉷 2011 John Wiley & Sons, Ltd.
