Laminar channel flow over long and moderate waves
E. D. Montalbano, M. J. McCready
Department of Chemical Engineering
University of Notre Dame
Abstract
Laminar Newtonian flow through a 2-dimensional channel with one sinusoidal wall is studied for
wavelengths comparable to, or much larger than, the channel height. In the long wave limit which
is solved analytically, the pressure is in phase with the wave slope and the shear stress is in phase
with the wave height. As the wavelength is reduced, the shear stress maximum moves upstream of
the peak and the pressure minimum moves toward the wave crest. Weakly-nonlinear effects
increase the shear on the upstream side of the crest and cause a net increase in the average shear
and pressure drop over the length of the wave. For moderate wavelengths the pressure drop and
drag are found, from finite element calculations, to increase as amplitude to approximately the third
power. The maximum shear stress moves upstream of the crest as Reynolds number is increased
and the phase angle decreases as the amplitude is increased. The minimum pressure moves
downstream, away from the inviscid value, as the amplitude is increased.
Version: 6/16/98
Keywords: wavy surface flow, finite element, longwave analysis, weakly-nonlinear analysis
2
1. Introduction
While turbulent flow over small and large amplitude wavy surfaces has been studied
extensively, (e.g. Buckles et al.,1984; Zilker et al., 1977; Kuzan et al., 1989) because of its
application to gas-liquid flows and as a test for turbulence models, laminar flow over wavy
surfaces has received much less attention. However, core annular flow, (Petziosi et al., 1990),
pipeline transport of settled solids (Kuru et al., 1995) and oil-water channel flows (Kim and Park,
1970) are examples where a laminar flow of a less viscous phase causes wave generation in a more
viscous phase. This process is driven by the interfacial shear stress and pressure variations, thus
knowledge of these components is crucial. Kuru et al. (1995a) have shown that linear stability
behavior of two-layer flows can be predicted reasonably well by using a divided attack (Benjamin,
1959), where flow of the less viscous phase is solved for a solid wavy surface and the stress
variations are imposed as boundary conditions on the less viscous phase, if the viscosity ratio is
greater than about 5. However while Patel et al. (1988) describe a procedure to obtain a numerical
solution to laminar channel flow over a wavy wall, they do not provide any results that are useful
in the wave generation problem. Furthermore, even when a weakly-nonlinear amplitude expansion
of full Navier-Stokes equations is used to solve for the amplitude of steady waves in a two-layer
flow (Renardy and Renardy, 1992; Sangalli et al, 1995), it is not possible to reliably calculate
pressure and shear stress components.
In this paper laminar channel flow of a Newtonian liquid over a solid wavy wall is
examined for small and order one channel height to wavelength ratios with the intention of
producing useable results for the wave generation problem in two-phase flows. The long wave
regime is studied analytically using linear and nonlinear theory and analytical expressions for the
shear and pressure variations are produced.
The moderate wavelength problem is solved
numerically with a finite element formulation providing qualitative trends and quantitative results.
3
2. Mathematical Formulation For Linear and Nonlinear Analysis
Linear analysis
The flow configuration is a channel with a wavy lower wall as shown in figure 1. The
relevant parameters for this flow are the Reynolds number, UH/ν, and the ratios
a/λ and
dimensionless wave number α=2πh/λ. The term a is the amplitude, λ is the wavelength, U is the
average velocity and H is the mean channel height.
The x and y coordinates have been
nondimensionalized with H and the velocities with U. Yih (1969) has given detailed derivation of
the governing equations for linear stability. If flow through a flat walled channel is disturbed by
the presence of a wavy wall, the linearized Navier-Stokes equations with a normal mode
assumption for the perturbed stream function, ψ'(x,y),
ψ '(x,y) = φ(y) exp( I α x),
[1]
and other variables yields the Orr-Sommerfeld equation with wave speed c=0,
φ iv − 2 α 2 φ ' ' + α 4φ = iα Re[U 0 ( φ ' ' − α 2 φ ) − U 0' ' φ ].
[2]
The boundary conditions are,
φ = φ ' = 0, at y = 1,
[3]
φ = φ ' = 0, at y = η (x),
where η(x)= a cos(αx) is the wavy surface and the primes denote differentiation with respect to y.
The undisturbed base state velocity profile U0(y) for a flat walled channel as is expressed
nondimensionally as
U o(y) = 6 y - 6 y2.
[4]
Note that in the normal mode assumption the complex exponential part is the functional form of the
wavy surface.
For the case, H<<λ and α->0, this problem can be solved as a regular perturbation in α.
The amplitude function of the disturbance stream function is written as,
4
φ ~ φ 0 + αφ 1 + α 2 φ 2 + ⋅ ⋅ ⋅
.
[5]
When this expansion is substituted into the Orr-Sommerfeld equation and boundary conditions and
terms of like order in α are grouped together, a simplified set of fourth order equations and
boundary conditions for φi result. These equations may be solved sequentially to determine φi at
each order of the approximation.
Application of the boundary conditions at the top of the channel is straight forward because
they are evaluated at a constant value of 1. To account for the wavy surface without the use of a
coordinate transformation, domain perturbation is performed on the boundary conditions. This
gives
φ ' ' ( y = 0) = − U 0 ( y = 0).
[6]
In the presence of a wavy boundary we expect a periodic variation of the velocity field,
pressure and shear stress at the wall. For a linear response,
_
τ w = τ w +a ^τ w cos(αx + θ)
[7 ]
and likewise for the pressure. Here a ^
τ w is the amplitude of the wave induced stress variation at
_
the wall, θ is the angle by which the maximum precedes the wave crest, and τ w is the shear for the
undisturbed flow. A Taylor series about y=0 of the total shear response and evaluated at y=a
cos(αx) gives the final form of the shear and the pressure at the wavy surface. Recalling equation
(1) this gives,
τw | y=a exp(Iαx)
=
∂3ψ
∂ 2φ 
1  ∂ 2ψ
+
a
exp(
I
α
x
)
+
a
exp(
I
α
x
)


Re  ∂y 2
∂y 3
∂y 2 
,
[8 ]
5
_
where the derivatives with respect to y are evaluated at y=0, and ψ is the stream function for the
undisturbed flow.
Weakly Nonlinear Analysis
Hooper and Grimshaw (1985) used weakly-nonlinear analysis to examine the nonlinear
instability of two layer Couette and Poiseuille flow under long wave conditions. This method has
often been used to derive amplitude evolution equations for interfacial waves. In the present work,
there is single fluid and a solid wave so that there are no interfacial boundary conditions. The
variables are nondimensionalized as in the linear analysis; the base state velocity is given by [4] and
the base state stream function is simply the integral of [4]. The base state is disturbed so that the
stream function equation is
y
ψ (x,y) = ∫ U0 ( y)dy + ψ ' ( x, y)
[9]
The stream function, ψ (x,y) satisfies the stream function form of the Navier-Stokes equations and
the disturbances of ψ satisfy
(U 0 (y) +
∂ψ ' ∂
∂ψ ' ∂ 2 U 0 ∂
) (∇2 ψ ' ) −
(
+ (∇2 ψ ' ))
∂y ∂x
∂ x ∂ y2 ∂ x
1 4 ' .
∇ψ
=
Re
[10]
The no slip conditions on solid boundaries require that
ψ = 0 at y = η = a cos (αx), ψ = Q * at y = 1.
and
[11]
6
∂ψ
∂ψ
=
0
and
∂y
∂ x = 0 at y = 1, η = a cos (αx),
[12]
where η is the surface of the wave and Q * is the dimensionless flow rate. Note that the surface
has not been expressed in complex form because doing so does not provide any simplification in
the solution procedure.
For a weakly nonlinear long wave analysis it is assumed that
η = a cos(α ξ ),
[13]
where stretching with respect to a in the x direction has been applied:
ξ = a x.
[14]
The disturbed stream function is then expanded in a perturbation expansion in powers of a
ψ'(x,y) = aψ0(ξ,y) + a2ψ1'(ξ,y)
+ a3ψ2'(ξ,y) + ... .
[15]
Equation (10) and the boundary conditions (11) are rewritten in terms of the new variables ξ and η
, i.e., ∂/∂x =
a ∂/∂ξ. Applying expansion (15) to the disturbed equation and its boundary
conditions and grouping terms of like order in amplitude yields a problem that can be solved at
orders
a1 and a2. As in the linear analysis, domain perturbation is used so that boundary
conditions at y=0 can be applied.
At each order, the equations are solved by guessing of the functional form of the solution.
At order a1, ψο is
ψ o (X,y) = A(X) φ ο (y) .
At order a2 the functional form is
(16)
7
ψ 1 (X,y) = A x φ 11 (y) + A 2 φ 12 (y) ,
which satisfies the forcing term AX
(17)
in the inhomogeneous equation and the A 2 term of the
boundary condition that occurs at this order. For more details see Hooper and Grimshaw (1985).
Numerical Formulation
The Navier Stokes equations are discretized in space with the penalty formulation Galerkin
Finite Element Method (PGFEM). The Penalty formulation consists of replacing the divergence
free constraint for the velocity, ui,i =0, with the perturbed equation ui,i = -εp, where p is pressure.
After discretization the perturbed equation is then solved for pressure and substituted into the discretized x and y momentum equations. Once the velocities are obtained the pressure is then
recovered from the perturbed continuity equation. This method has been found to be equal in
accuracy to the v-p formulation (S. W. Kim, R. A. Decker (1988)).
Nine node biquadratic basis functions were employed for the velocity unknowns and three
node linear discontinuous basis functions for the pressure.
The velocity nodes were defined
globally while the pressure nodes were located at the integration points for 3 point Gaussian
integration within the parent or master element. This 9/3 element has been found to be highly
accurate and stable by Oden and Jacquotte (1984).
In the flows solved here, the domain is open ended. Gresho (1991) shows that while not
physically meaningful, the boundary conditions from the conventional form of the viscous terms
are very useful and in a mathematical sense more reliable at exit flows.
Thus for numerical
solutions presented in this paper, the conventional form of the viscous terms is used. This form of
the viscous terms allows for exit Neumann boundary conditions to be zero. In all runs the domain
is long enough so that end effects can be neglected and the solution is obtained from one wave in
8
the center of the domain. If a better discretization is desired, solutions from the coarse solutions
are used as boundary conditions to solve the more refined problem over one wave.
An implicit time factorization scheme is used to integrate the semi-discrete Navier Stokes
equations. This method is second degree accurate and provides a non-iterative model to the
nonlinear problem. In most cases presented here the steady state solution was of main interest and
was obtained using a direct iteration with an over relaxation parameter to ensure convergence (Kim
and Decker (1988)).
More details of the formulation and implementation of the numerical scheme are given in a
thesis by Montalbano (1994). The correctness of the code was tested by comparison with the
analytical solutions to transient Couette flow and an oscillatory flat plate in a finite domain. In both
cases, the analytical and numerical solutions agree. The results for linear Stokes flow are in perfect
agreement with other published results (Hughes et al. (1979), Bercovier and Engelman (1979),
Soh and Goodrich (1988)) Those for the high Reynolds number problem show an excellent match
to other workers (Kim and Decker (1989), Ghia et al. (1982), Schreiber and Keller (1983)) when
comparing center line velocity profiles and location and values of the primary vortices.
Analytical and Numerical Results
Results for the Linear Long Wave Analysis
τ w from equation (7) in complex form is
The perturbation shear stress ^
^τ w
12
6 I
128
4 R e
 24

6 I
= Re + α  + 35 R e + α 2 
R e + 5 Re + 2695 
Re

 35

128
4 Re
17 I
8 I

α3
R e + 5 Re + 2695 - 943250 Re 2  + O(α4)
 35

The preturbation pressure is
+
[18]
9
^p w =
36 I
α Re +
 327
α2  175
3 6 I
 54
+
R e 
 35
216 I
117 I
 54

+ α - 35 + 5 Re + 13475 R e +


2
216 I
117 I
1233 Re 
+ 5 Re + 13475 Re + 12262250  + O(α3)

[19]
It is seen that as α->0, the leading order term of the shear is in phase with the crest of the wave.
The second term causes the maximum in shear stress to move upstream of the crest. For pressure,
the leading order term is in phase with the wave slope.
The magnitude of the shear stress
fluctuation increases with increasing wave number as does the phase shift of the shear. For the
pressure however, the magnitude of the fluctuation decreases and the phase shift increases with
increasing wave number. The effect of increasing Re is to make the magnitude of the variation
larger. Also since the linear approximation uses a normal mode assumption for the disturbance,
the results consist of a response that is symmetric in the x direction.
Results for the Nonlinear Long Wave Analysis
In the weakly nonlinear analysis, the wavy surface is defined in the same way as the linear
case but the solution allows for inclusion of additional modes. This enables determination of the
nonlinear effects on the shear and pressure along the surface of a wave. Also, the response will be
nonsymmetric. The amplitude, wave number and Reynolds number determine the effect of the
nonlinear contribution.
The equation for shear obtained by nonlinear analysis is
^τ w A[x]
12
6
72
= Re A[x] + 35 A[x]x + Re A[x]2 + O(α2,a)
and for the pressure
[20]
10
-p w
<p w>
-12
sin(αx) 54
54 sin(2αx)
= R e x + a(36 α Re - 35 cos(αx)) + a2( 2 α Re
54
- Re x) + ... .
[21]
Figure 2 displays the total shear variation from Eq. 20, as well as the contribution from
linear and nonlinear terms of the expansion for Re=500,
a /H = 0.05 and α = 0.02, 0.05, 0.08.
The nonlinear contribution, which consists of the overtone of the wave and a constant term, is
positive over most of the wave; its largest contributions occur a little downstream of the crest and
the trough. The effect is more pronounced as the wavenumber increases. Nonlinear effects
increase the maximum in stress, which is upstream of the crest for the linear case, and moves it
back toward the wave crest. The nonlinear correction reduces the drag in the trough and moves the
maximum negative value upstream. There is an overall increase in drag because of the presence of
the 3a2 term. The major nonlinear correction to the pressure variation is the additional pressure
drop term, -54/Re x, which increases the pressure drop substantially. The other nonlinear term
contributes only slightly to the result.
Strongly Nonlinear Problem - Numerical Results
The Navier-Stokes equations were solved for velocity and pressure using the finite element
method. The lid height was maintained at 1.0 for all runs. For dimensionless wave numbers of
0.754 and 0.377, runs were performed with
a =.125, .25, .375, and .5, for Re=10, 50, 75, 100,
200, 300, 400, and 500. These wavelengths and amplitudes are out of the range of validity of the
analytical analysis.
Pressure Drop
Figure 3 shows the deviation of mean pressure drop compared to a flat walled channel
(superscript "s" denotes a smooth channel) with Reynolds number for different amplitudes. At
11
low Reynolds number the slope is small. Starting at Re=75 the slope increases. This is typical of
flow through constricted tubes and is caused by inertia beginning to dominate the problem at higher
Re. The most dramatic result is the increase in pressure deviation with the increase in amplitude
even at the smallest Re of 10. At a=.125 the pressure drop is 5% above that for a flat walled
channel and for
a=.5, it is about 150%. This effect scales with amplitude approximately as a 3 .
When comparing these results to those at α=.754, there was little if any difference between the two
wavelengths.
Drag
Figures 4 and 5 show the components of drag as a function of Reynolds number for
a
=.125 and a =.5 and a wavelength of 8.333 (α = .754). At smaller amplitudes, the viscous drag
on the lid and the wave are identical at Re less than 100 at which point they begin to deviate
slightly. The slope of the pressure drag remains constant up to Re=50, then the slope decreases.
The pressure drag is small and has little influence on the total drag at this amplitude. For
a=.5 the
drag on the lid and the wave deviate slightly at the smallest Reynolds number and at Re=50 the
difference begins to increase due to a decrease in the slope of the drag on the lid and an increase in
the slope for the drag on the wave. For this amplitude, the pressure drag is greater than the
viscous drag on the lid or the wave and it plays an equal part in the total drag. Comparing Figure 4
to 5 the pressure drag has increased approximately 30 times going from
a=.125 to .5 and the
viscous drag for the lid and the wave by a factor of approximately 1.30. A trend shown here as
well as by Patel (1991a,1991b) is that the viscous drag on the lid is greater than the drag on the
wave. Also in each plot, drag reduction is present for those Re in which recirculation occurs due
to inertia causing the bulk of flow to bypass the troughs.
Phase Shift
Figure 6 shows the shear stress phase angle (degrees upstream of the crest) as a function of
Reynolds number for all four amplitudes with λ=16.666 (α = .377). At all amplitudes there is a
sharp increase in phase at low Re which then levels off at around Re=300. Also as amplitude
12
increases there is a decrease in phase in shift that is close to linear. At low Reynolds numbers,
both Re and amplitude are important for determining phase shift; at higher Reynolds numbers
amplitude is the determining factor.
Equation 18 shows that the phase shift of the shear increases with increasing wave number
for long waves. Numerical results of Montalbano (1994) (not shown here) agree with the long
wave trends up to dimensionless wave numbers of 0.75. For wave numbers larger than .75, the
phase shift decreases with increasing wave number in agreement with Benjamin (1959) who
analytically demonstrated this decrease with increasing α.
The phase angle for the pressure minimum varies from -1020 to -1400 as the Reynolds
number increases from 10 to 500. The inviscid value is -1800. For fixed Reynolds number the
phase angle moves away from the inviscid value as the amplitude is increased. For Re = 100, the
phase angle is -1370, -1310 and -1010 for amplitudes of 0.125, 0.25 and 0.5. Similar behavior
occurs at other Reynolds numbers. The effect is even larger if separation occurs.
Profile Of Shear And Pressure Along Wave Surface
Profiles of the shear stress and pressure along the wavy surface for Re=500 and
a=.125
and .5 are shown in figures 7 and 8 respectively. Plots of the pressure minus the average pressure
are also given to show the periodic behavior of the pressure. In figure 7, the shear stress and
pressure are shifted substantially from their limiting values even though the Reynolds number is
high. The components display nearly linear behavior with a slight distortion of the sinuous shape.
There is a slight increase in the shear on the windward of the wave compared to the leeward side.
The shape of the shear profile near the peak is sharp and the increase at the peak is greater in
magnitude than the decrease in the trough. This agrees with weakly nonlinear analysis.
larger amplitude in figure 8,
For the
a=.5 and there is a large region of recirculation in the trough. The
ordinate (τ-τavg)/τavg = -1 (i.e., τ = 0) at the attachment and reattachment points. The pressure
phase shift is less for the larger amplitude. There is a noticeable pressure change at the separation
point and a recovery at the reattachment point.
13
Discussion
The results presented here for fluid flow over a solid wave are expected to provide insight
into two-layer flow problems where the viscosity ratio is very far from unity. Kuru et al. (1994a)
compare results for the two-layer stability problem using the full boundary conditions and the OrrSommerfeld equation for each phase with the "divided attack" where the interfacial stress variations
are calculated for the flow of the less viscous phase over a solid surface and then used as boundary
conditions on the stability problem for the lower phase. For viscosity ratios less than 1/5, they
find that the long wave results agree very closely. For moderate wavelength waves, there is a
systematic error, but the qualitative behavior is similar.
The behavior of shear and pressure variations for long wavelength waves is quite different
from unbounded systems where the waves are short. The maximum pressure for long waves
occurs 900 downstream of the tough compared to the high Re, short wave value which would be in
the center of the trough. This result is important because extrapolated data from shortwave channel
flows (e.g., Thorsness et al. (1978)) suggest that the pressure minimum remains near the crest as
the wavenumber goes to 0. Note that for long waves, turbulent flow is expected to behave
qualitatively similar to laminar flow because there are no high curvature regions where the dynamic
effects of turbulence should be important.
The nonlinear behavior for long waves suggests that it is not likely that linear processes
such as formation of waves on gas-liquid flows or "dunes" in flows over settled particulates
(Takahashi, et al. 1989) will be saturated or diminished as the amplitude increases due to effects in
the low viscosity phase. The weakly nonlinear effect is primarily to increase shear near the crest;
the effects on the phase angles are minor. Thus any mitigation of the growth of disturbances must
come from changes within the more viscous phase. The picture could be different for larger
14
wavenumber. Figure 6 shows that the phase angle for the shear stress decreases significantly as
the amplitude increases. Thus phenomena such as waves on very thin liquid layers (Craik (1966))
which rely on an out of phase shear stress (i.e. in phase with the wave slope) to cause the
instability, could have their growth arrested by nonlinear effects. Waves or dunes that form on
settled particulates may be affected as well because the waves travel by erosion on the windward
slope (Kuru et al. (1994b)). However, the minimum pressure moves further from the crest as the
amplitude increases suggesting that no reduction in the growth of waves in gas-liquid flows, which
are initially moderate to short wavelength, is expected. The pressure component in phase with the
wave slope is responsible for this instability (Hanratty (1983)).
Acknowledgments
This work has been supported by the Department of Energy, Office of Basic Energy Sciences
under grant DE-FG02-ER-13913. The authors thank Dr. J. F. Brennecke for helpful discussions.
References
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15
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for high-Reynolds number flows’, Int. J. Num. Meth. Fluids, 6, 25-35 (1988).
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16
Figure 1.
Geometry of Channel
Figure 2.
Shear stress deviation over a wave comparing linear and nonlinear
contributions Re=500, α=.02, .05, .08
Figure 3.
Variation of pressure drop deviation from that of a flat walled channel with
Reynolds number of several amplitudes. α=.754.
Figure 4.
Variation of components of drag with Reynolds number.
Wave length=8.333, Amplitude .25
Figure 5.
Variation of components of drag with Reynolds number.
Wave length=8.333, Amplitude =.5
Figure 6.
Variation of phase shift with Reynolds number for a=.125, .25, .375, .5
for λ=16.666.
Figure 7.
Top: profile of shear (dotted) along wave surface.
Middle: profile of pressure less average pressure (dotted) along wave
surface.
Bottom: profile of pressure (dotted) along wave surface.
Solid curves are the surface of the wave. Re=500 a=.125 α=.753
Figure 8.
Flow separation occurs between x/λ-0.1 and 0.52
Top: profile of shear (dotted) along wave surface.
Middle: profile of pressure less average pressure (dotted) along wave
surface.
Bottom: profile of pressure (dotted) along wave surface.
solid curves are the surface of the wave. Re=500 a-.5 α=.753
17
Channel Lid
Wavelength λ
Solid Sinuous Wave
Amplitude
Figure 1. Geometry of Channel
3
18
surface of wave
total disturbance
linear contribution
nonlinear contribution
0.10
τ/τavg-1
0.05
0.00
-0.05
α = 0.02
-0.10
0.0
0.2
0.4
0.6
0.8
1.0
X/λ
0.10
τ/τavg-1
0.05
0.00
-0.05
α = 0.05
-0.10
0.0
0.2
0.4
0.6
0.8
1.0
X/λ
0.10
τ/τavg-1
0.05
0.00
-0.05
α = 0.08
-0.10
0.0
0.2
0.4
0.6
0.8
1.0
X/λ
Figure 2. Shear stress deviation over a wave comparing linear and nonlinear contributions.
Re=500, α=.02, .05, .08
19
4
1
8
6
4
s
[∆P/L-(∆P/L) ]/(∆P/L)
s
2
2
0.1
8
6
a=.125
a=.25
a=.375
a=.5
4
2
0.01
2
10
3
4
5
6
7
8
9
2
3
4
5
100
Reynolds Number
Figure 3.
Variation of pressure drop deviation from that of a flat walled channel,
(∆P/L)s, with Reynolds number for several amplitudes. α=.754.
20
4
Viscous Drag on Wave
Viscous Drag on Lid
Pressure Drag on Wave
Viscous Drag for a=0
2
1
Drag
6
4
2
0.1
6
4
2
0.01
2
3
4
5
6
7
8
9
10
2
3
4
5
100
Reynolds Number
Figure 4. Variation of components of drag with Reynolds number.
wavelength=8.333, amplitude=.25
4
2
1
Drag
6
4
2
0.1
6
4
2
0.01
2
10
3
4
5
6
7
8
9
2
3
100
Reynolds Number
Figure 5. Variation of components of drag with Reynolds number.
wavelength=8.333, amplitude=.5
4
5
21
Phase angle (degrees) for Shear
30
25
20
15
10
a= 0.125
a= 0.25
a= 0.375
a= 0.5
5
0
0
100
200
300
400
Reynolds Number
Figure 4. Variation of phase shift with Reynolds number for a=.125, .25, .375, .5
for λ=16.666
500
22
0.8
0.10
0.6
Surface
0.2
0.00
0.0
-0.2
-0.05
(τ−τavg)/τavg
0.4
0.05
-0.4
-0.10
0.0
-0.6
0.2
0.4
0.6
0.8
1.0
0.10
0.20
0.05
0.15
0.00
0.10
-0.05
0.05
-0.10
0.00
0.0
0.2
0.4
0.6
0.8
(P-Pavg)
Surface
X/λ
1.0
X/λ
0.10
0.20
0.15
0.10
0.00
0.05
Pressure
Surface
0.05
0.00
-0.05
-0.05
-0.10
-0.10
0.0
0.2
0.4
0.6
0.8
1.0
X/λ
Figure 7. Top: profile of shear (dotted) along wave surface.
Middle: profile of pressure less average pressure (dotted) along
wave surface.
Bottom: profile of pressure (dotted) along wave surface.
Solid curves are the surface of the wave. Re=500 a=.125 α=.753
0.4
6
0.2
4
2
0.0
x/λ = .1
x/λ = .52
0
-0.2
-2
-0.4
0.0
(τ−τavg)/τavg
Surface
23
0.2
0.4
0.6
0.8
1.0
X/λ
1.4
0.4
1.2
1.0
Surface
0.8
0.0
0.6
0.4
(P-Pavg)
0.2
-0.2
0.2
0.0
-0.4
0.0
0.2
0.4
0.6
0.8
1.0
X/λ
-3.2
0.4
-3.4
Surface
-3.6
-3.8
0.0
-4.0
-4.2
-0.2
-4.4
-0.4
0.0
-4.6
0.2
0.4
0.6
0.8
1.0
X/λ
Figure 8.
Flow separation occurs between x/λ = 0.1 and 0.52
Top: profile of shear (dotted) along wave surface.
Middle: profile of pressure less average pressure (dotted) along
wave surface.
Bottom: profile of pressure (dotted) along wave surface.
Solid curves are the surface of the wave. Re=500 a=.5 α=.753
Pressure
0.2