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Stratified flow past a bell-shaped hill
Fluid Dynamics Research 9 (1992) 1-18 North-Holland
FLU I D D ? N A M I C S RESEARCH
Stratified flow past a bell-shaped hill Masahiro Suzuki Institute of Computational Fluid Dynamics, 1-16-5 Haramachi, Meguro-ku, Tokyo 152, Japan
and Kunio Kuwahara The Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara-shi, Kanagawa 229, Japan
Received 9 March 1990 Abstract. A numerical study of the stratified flow past a bell-shaped hill is presented. The three-dimensional unsteady Navier-Stokes equations under the Boussinesq approximation are solved by a finite difference method in a generalized coordinate system. The characteristic phenomena of the stratified flow with various Froude numbers are successfullysimulated. The results are compared with theoretical and experimental data and good agreement is obtained.
1. Introduction
Stratified flow around surface-mounted hills is a subject that has attracted great interest from not only fluid dynamicists but also engineers in m a n y fields. It has strong relationships with industrial and environmental problems, as follows: (a) Air pollution problems, such as pollutant transport and dispersion over complex terrain (Snyder, 1985). (b) The prediction of the wake flow over mountains for aviation safety. The rotors behind mountains caused serious air accidents in the past. (c) The study of air flow over a complex relief necessary for effective wind energy utilization (Hunt et al., 1984). (d) Meteorological problems, for example, von Kfirmfin vortex street in the wake of an isolated island (Tsuchiya, 1969). An enormous number of investigations have been mainly conducted by the following four methods: (1) field study, (2) theoretical analysis, (3) experimental study, and (4) numerical study. Field study has contributed to discovery and understanding of the p h e n o m e n a of stratified flow since the 1920s, when glider pilots in the Alps discovered lee-waves (Lugt, 1983). In the 1960s the von Kfirm~n street-like vortex, which trails behind the islands in the earth's atmosphere, was discovered by a new tool of field study, i.e. meteorological satellites (Berger and Wille, 1972). ' Though many theoretical studies for two-dimensional flows have been conducted (Queney, 1948; Long, 1957), only a small number of results have been obtained for three-dimensional flows. For three-dimensional flows, linear perturbation analyses have been studied (Crapper, 1959; Umeki and Kambe, 1989), but nonlinear theoretical results, which can be applied to a 0169-5983/92/$5.50 © 1992 - The Japan Society of Fluid Mechanics. All rights reserved
2
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
.
.
.
.
.
~Fh
Fig. l. Small-Froude-number theory for gcr~,~;)vera three-dimensional hill by Drazin: Definition of regions (from Hunt and Snyder, 1980).
three-dimensional hill that is not a small perturbation of a plane surface, was obtained only by Drazin (1961). Drazin assumed as a first approximation that the flow is confined to horizontal planes and then developed a solution as a series of the small Froude number F. Brighton (1978) extended the solution to a higher order of the Froude number. For strongly stratified flow past obstacles, Sheppard (1956) proposed the formula
pu2( z = U p ) = g f o( h - z ) ( - d p / d z
) dz,
(1)
which is derived from the relation between the kinetic energy of the parcel, whose far upstream at elevation is HD, and the potential energy gained by it when being lifted through the density gradient. In eq. (1), U(z) is the approaching velocity, p(z) is the density of the fluid, h is the height of the obstacle, and g is the gravitational acceleration. If the fluid parcel is originally at a higher level than H D , it goes over the top of the obstacles, otherwise it may go around the sides. H D is called the "dividing-streamline height", which is a very important quantity for the assessment of pollutant concentration around mountains. Experimental approaches have been successful in explaining some of general phenomena. Hunt and Snyder (1980) (hereafter referred to as HS) conducted towing-tank experiments for a three-dimensional bell-shaped hill in a linearly stratified environment with an effectively uniform approaching flow. They verified that Drazin's theory is applicable over the Froude number range of F < 0.4, i.e. (1) The fluid moves approximately in the horizontal direction in the two regions H 1 and H 2. (See fig. 1 for the nomenclature.) (2) The flow is not restricted to move horizontally in regions B and T, whose thicknesses are about Fh. (3) In the region Hi, there exists mainly horizontal recirculation flow. They also investigated the dividing-streamline height and suggested the simple formula Hr, = h(1 - F ) ,
(2)
for 0 < F < 1. This formula can be derived from Sheppard's formula under the condition of uniform velocity and linear stratification. Unlike Sheppard's original formula, this does not have to be solved iteratively for the unknown H D. Snyder et al. (1985) verified Sheppard's formula for a variety of shapes with different upwind density and velocity profiles. They concluded that the formula provides a good estimate for H D for symmetric hills and small upwind shear. Brighton (1978) carried out experiments for a hemisphere, a cone, and a truncated cylinder in a closed-circuit stratified water channel. He observed that vortices are shed in region H t for F < 0.15 irrespective of the body shape and the Reynolds number (Re > 500), and the shedding frequency is the same at all heights. His experiments also showed a cowhorn-shaped eddy, which has a horizontal axis in the lee at the level of the top of the obstacle. He attributed its generation mechanism to the intensification o f . t h e eddy due to vortex stretching. A strong vertical shear exists where fluid, which clears the top of the obstacle
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
3
and separates from its lee slopes, flows over the stagnant horizontal recirculating flow region. A "multiple bubble", which is generated by an interaction between the flow of region T and the separated wake of region H1, and the lee wave crest together act as an effective blockage. They cause vortex stretching and the resulting vortex intensification. The vorticity is swept downstream in the trailing arms of the cowhorn eddy and is soon suppressed. But Castro et al. (1983) did not observe the cowhorn eddies in their experimental studies. They reported that vortex shedding is possible even at F = 0.3. Owing to the recent considerable advance in supercomputers, numerical studies have become powerful approaches compared with experiments. However, few numerical computations of three-dimensional stratified flow have been conducted. Sykes (1978) computed laminar flow over two-dimensional hills only. Ruscher and Deardorff (1982) applied a two-dimensional model of the atmospheric planetary boundary layer to simulating the development of a Kfirmfin vortex street downstream of the island of Cheju-do. Smolarkiewicz and Rotunno (1989) investigated the flow of a density-stratified fluid past a three-dimensional obstacle, using an inviscid, anelastic, nonhydrostatic finite difference model. Hanazaki (1988) conducted three-dimensional computation of viscous incompressible non-diffusive stratified flow past a sphere at Re = 200. But there have been no numerical studies on three-dimensional flow of higher Reynolds number. In this study we simulate stratified flow past a bell-shaped hill at high Reynolds numbers using the three-dimensional unsteady Navier-Stokes equations under the Boussinesq approximation. The governing equations are solved by a finite difference method using a generalized coordinate system. The results are compared with theoretical and experimental data. Our computations are conducted under the conditions considered by HS, which serve as reference data, and the three-dimensional structure of the wake is extensively investigated. 2. Numerical model and procedures
2.1. Model We consider a flow with vertical temperature gradients passing a bell-shaped obstacle on a fiat plate. The bell shape of the obstacle is defined as
f ( r ) = 1 / ( 1 + r4),
(3)
where f(r) is the height and r is the distance from the center of the obstacle. This shape is slightly different from the model used by HS. The approaching flow has stable and linear density gradients. No turbulence models are employed. In this study, the effect of the rotation of the earth is ignored, because the Rossby number is sufficiently large if the base diameter of the hill is less than approximately 5 km (Snyder, 1981).
2.2. Governing equations Under the Boussinesq approximation the dimensional governing equations are written as follows: Continuity equation: (4)
div' u' = 0. Momentum equation: 0u'
~~t
+ (u"V')u'
1 ~, g po(Z = O) grad' + po(Z = O) V 1
po(z = O) ( P ' - Po)gZ'.
p2
u
t
(5)
4
M. Suzukg K. Kuwahara / Stratified flow past a bell-shaped hill
Energy equation: 0T'
~.
Ot-----7-+(u"V')T'
,2T,
O,o(z=O)CpV
.
(6)
Equation of state for the incompressible fluid:
p'= p'o(Z = 01[1 - a(T~;C To')]. ¢
t
(7) t
¢
t
p
Here, u' is the velocity (u;, u2, u3) [ = (u , v , w')], t' is the time, O is the density, po(Z) is the given density distribution of the approaching flow, T ' is the temperature, To'(Z ) is the temperature distribution corresponding to p'o(Z), ~a, is the perturbation pressure, # is the viscosity, g is the acceleration due to gravity, z' is the unit vector along z-axis, )t is the thermal conductivity, a is an expansion coefficient, Cp is the specific heat at constant pressure, and the primes indicate dimensional values. We non-dimensionalize velocities by the uniform approaching velocity U, distances by the height of a hill h, time by h/U, density by Oo(Z --- 0), perturbation pressure by po(Z = 0 ) U 2 and temperature by To'(Z = 0). Using eq. (7) and assuming a = 1/To'(Z = 0) (perfect gas), the following non-dimensional equations are obtained: Continuity equation: div u = O.
(8)
Momentum equation:
Ou
1
0-t- + ( u ' V ) u =
- g r a d ~ + -R-TV
2R__
1 ( r ° - T)z
(9)
~
Energy equation:
OT 0-7 + ( u ' V ) T =
1 Re. Pr v2T"
(10)
Here, Re is the Reynolds number [ = Uhpo(Z = 0)//z], F is the Froude number ( = U/Nh), N is the Brunt-Vaisal~i frequency, and Pr is the Prandtl number ( = Ce~/Oo(Z = 0)~). The values of the air under the standard atmosphere at 300 K are used for p0(0), /~, A and Cp.
2.3. Computational method The numerical scheme is based on the well-known MAC method (Harlow and Welch, 1965). First, the Poisson equation for the pressure, derived by taking the divergence of eq. (9), is solved at time step n:
v 2 ~ "= -div[(u"'V)u"] -
div[ ( T ° - Tn)z] + A t '
(11)
where D = div u, superscript n denotes the non-dimensional time t = nat and At is the time increment of finite difference approximation of 3/Ot. Though D in eq. (11) is identically zero due to eq. (8), it is retained here to prevent the accumulation of numerical errors and to ensure the continuity equation implicitly. By the substitution of eq. (8) into the first term at the right-hand side, eq. (11) is rewritten in the form 3
3 3u~' 3u~
V2ga"= - Z
E Oxj Oxi
i=1
j=l
1
Dn
F~div[(T 0 - r")z] + A-7'
where (x l, x 2, x3) [ = (x, y, z)] are the variables in the physical plane.
(12)
M. Suzuki, K. Kuwahara /Stratifiedflow past a bell-shaped hill
5
Secondly, u and T at the next time step are obtained by solving the Navier-Stokes and the energy equations, un+l
At
Un
+ ( u n ' V ) u " + 1 = - g r a d ~ n + ~_~ V2un+l_ ~F1 (To- T")z,
T n+l - T" At +(u"'V)Tn+I-
1 - - Pr V ZT'+l Re.
(13) (14)
The computations are done on a generalized coordinate system, which is obtained from the following coordinate transformations (Thompson, et al., 1985):
xa=x,(~', ~z, ~3),
x 2 = x 2 ( ~ , ' ~2, ~3),
x3=x3(~a, ~2, ~3),
(15)
where (~1, ~2, ~3) [= (~, ~/, ~)] are the coordinates in the transformed plane. Equations (12)-(14) are written in the generalized coordinates as
E ~ =-
g,j
+ .i
E E
/=1 j=l
1
ai"
-~
a J" ~
-~
-
i=lE ai"
(TO- T")Z]
3
+ ~ Y'. a i" Ou----~ i=1
(16)
O,f '
3 aiO(uk)n+l (."+l)k -- (.")k At + i=1 y'~ u"" a~ ~
= - E (i=1 a)k-~7-+-~E 1
Fz(To-T")(z)k T n+l - T"
At
- R
3
g'J O~ i ~)~Y.+ a'
0~' ]
(k=1,2,3),
(un+l)k
(17)
0T,+1
+ E un " ai i=l
Pr i=1 y'~ =
i=1 j E =l
~i
gij a~ i a ~ . + a i, a~' ! -gC rn + ~'
(18)
where a ~ ( = V~ ~) is the contravariant base vector, g~J ( = a ~, a j) is the contravariant metric tensor, and subscript k denotes the component in the k-direction. The base vectors are represented by the second-order central-differences scheme. All spatial derivatives, except nonlinear terms in (16)-(18), are approximated by the second-order central-differences scheme. The nonlinear terms are represented by a third-order upwind scheme (Kawamura, et al., 1986), .i.e.
(f~_~3u)i-Ji-fui+2-2ui+l+9ui-lOui6 A~i
1+2ui
2
=f~-2ui+2+lOu'+l-9U~+2u'-l-ui-2
for ~ > 0, for f / < 0 ,
(19)
6 A~i
where f is an arbitrary function and subscript i denotes the grid point. These difference equations are solved by a modified SOR (successive over-relaxation) method which contains no recursive references in the DO loop and achieves high vectorization on supercomputers.
6
M. Suzuki, K. Kuwahara / S t r a t i f i e d f l o w past a bell-shaped hill
2.4. Grid system In figure 2 the grid system is presented. The number of grid points distributed to each coordinate direction is 173, 89 and 63 in the ~, , / a n d ~"directions, respectively. The surface grid system on the hill and floor (~" = ~'0) is first formed, then the grid system along ~"= constant is heaped up in the ~'-direction (in this case 62 times) by a parabolic-hyperbolic hybrid grid generation scheme (Nakamura and Suzuki, 1987). [These grid surfaces of ~"= constant are not necessarily parallel to the plane of • = constant]. The grid points must cluster near the lower boundary to calculate the boundary layer accurately. The minimum spacing between two neighboring grid points is customarily fixed at 0.1/RvrR-e (Obayashi and Kuwahara, 1986). Therefore, the minimum spacing is set to be approximately 3 x 10 -4. In order to ascertain grid convergence, computations on a finer grid are tested. The method of grid doubling in each direction was proposed to ascertain grid convergence (Roache, 1989). However, increasing the number of grid points in each direction by a factor of 2, i.e. a total increase by a factor of 8, is not feasible with our computational facilities. Therefore, at this stage, a grid of 215 x 119 x 80, which is the maximum grid point number with our computer system, is used. No significant differences in the large structures of the flow field between results on the finer grid and these on the original grid are found. Though the test does not prove grid convergence, it confirms that a grid of 173 x 89 X 63 is fine enough to obtain the general features of the flow. A more accurate test remains a task for future computers.
2.5. Initial and boundary conditions We adapt the following initial condition as the impulsive start: u = (1, 0, 0), ~a= 0 and T = To(z ) in all regions. The boundary conditions are as follows.
'r
A p p r o a c h i n g flow
~"---~1
2 h
(a) 1Lli] ~ ~ ] i i t
111t i li~lll rlli i i ]
L+[~
~ ',,.;,,w,', ',4,,M~
zI
@
•",
,
, <~
,
,
~
Ib) Fig. 2. Grid system (173 x 89x63 grid points). (a) Hill surface and side, outflow boundaries, (b) side view of the centre-plane (y = 0 plane). [Note: the origin of the physical coordinate system (x, y, z) is located at the centre of the
hill].
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
7
On the outer boundary except upstream: u, ~ and T are determined by an extrapolation. On the upstream boundary: u = (1, 0, 0), ~ = 0 and T = To(z ). On the lower boundary: u = (0, 0, 0), while ~ and T are determined by the extrapolation. The outer boundary condition ensures the condition of infinite boundaries. According to the linear theory (Janowitz, 1981), the columnar disturbance becomes weaker as h / D ( D is the half-depth of the channel) becomes small and, finally, is expected to disappear at D ~ oo (infinite upper boundary). Though the upstream influence is observed in the stratified flow (Baines, 1987), the upstream condition is fixed to be uniform. Because of the remoteness of the upstream boundary from the hill and also because of the rough mesh in the upstream region, the upstream influence will be damped before reaching the upstream boundary. To identify the flow variations with Froude numbers, the cases of F = co, 1.5, 1.0, 0.5, 0.2, and 0.1 are calculated at Re = 105. Computations are mainly done on the Fujitsu VP-400E, which is a 1700 Mflops supercomputer at the Institute of Computational Fluid Dynamics. Typical CPU time is about 10-20 h for each case.
3. Results and discussion
We mostly observed quasi-stationary flow patterns in stably stratified flow after non-dimensional time t --- 25 except for vortex shedding in the horizontal recirculating region. Thus, some of the typical instantaneous flow patterns are presented here. 3.1. The dividing-streamline height
Computed particle traces emerging from upstream at several levels in the centre-plane ( y = 0 plane) are obtained to examine dividing-streamline heights for 0 < F < 1. The results are plotted in fig. 3. Because the approaching flow has a uniform velocity at the upstream boundary and the no-slip condition is applied on the lower boundary, the boundary layer may develop depending on the Reynolds and the Froude numbers as the flow reaches the hill. In the absence of a hill, i.e. a flat plate, the flow in the boundary layer is characterized by the Reynolds number Re~ with reference to the distance from the upstream edge of the flat plate. Since the distance from the upstream boundary to the centre of the hill is 12h and the Reynolds number Re based on the hill height is 105 , Re, = U12hpo(Z = 0)//z = 12 Re = 1.2
x
106.
This is less than the critical value of Re~ for the onset of turbulence in the laminar boundary
1.0
0.5
~, \\ \\
i \\
o.
o.5
"\\
\\
\
1.0 F Fig. 3. Dividing-streamlineheight versus Froude number (Re ~ 105). The drawn line shows HD/h = 1 F.
8
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
layer on a flat plate (Hinze, 1975). Thus, the flow in the boundary layer can be expected to be laminar at the centre of the hill (but in the absence of it). According to Blasius' solution for the plane laminar boundary layer, the boundary layer thickness 8, at which the velocity equals 99.7% of the constant free-stream velocity, can be obtained: 8 = 5.5 x
12h/R~x= 0.06h. +++.~+
i
- - ~ ~ + +
-'+~-~~'---~-~--
....
~ +
Fig. 4. Surface streamlines in the centre-plane ( y = 0 plane) a n d surface shear-stress p a t t e r n s o n the lower b o u n d a r y (Re = 105, t = 30), (a) F = 1.0, (b) F = 0.5, (c) F = 0.2. The origin of the axes (x, y ) is located ai the centre of the hill in the surface shear-stress pattern.
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
Fig. 4 (continued).
The computations, which are done on a grid with the same distribution of points but in the absence of the hill, show approximately the same degree of boundary layer thickness. No significant variations of the boundary layer thickness with the Froude numbers are found. Therefore the boundary layer is thin enough, and the present results are in good agreement with HS' formula H D = h(1 - F), which is valid for effectively uniformity approaching flows. Slight deviations at a higher Froude numbers ( F = 0.5 and 1.0) are mainly due to the existence of horseshoe vortices (arrows in figs. 4a and 4b), which produce some downwash on the upwind slope of the hill (Castro et al., 1983). 3.2. The lee-wave structure
The surface streamlines in the centre-plane ( y = 0 plane) and the surface shear-stress patterns on the lower boundary at F = 1.0, 0.5 and 0.2 are shown in fig. 4. At a Froude number of 1.0, the fluid flows along the lee slope without separating. Separation begins farther downstream of the hill. At a Froude number of 0.5, separation begins roughly half-way down the lee slope. There are reverse flows under the lee-wave. The amplitude of the lee-wave decreases rapidly in the flow direction. Some of the streamlines do not go over the top of the hill. At a Froude number of 0.2, the flow separates just when it passes the top of the hill. The range of heights at which the streamlines go over the top of the hill shrinks. The vertical movement of the flow is suppressed. Though the locations of separation lines, which are observed in the surface shear-stress patterns, coincide with HS' results, the surface shear-stress patterns in the wake are different from those of the HS. The patterns are more complex. No symmetric pair of prominent singular points is clearly found in the wake. There are many small singular points. Since HS drew the surface shear-stress patterns by collating all observations, the patterns are somewhat speculative mean patterns. Accordingly, the differences in the surface shear-stress patterns between HS and the present computations are mainly due to the difference between instantaneous and time averaged flow patterns.
M. Suzuki, K. Kuwahara
10
/Stratifiedflowpast
a bell-shaped hill
Table 1 Comparison of theoretical and observed lee-wavelength ?~ Froude number
X/h
theory 1.0 0.5 0.2
observed
6.28 3.14 1.26 ~,~,
6.0 2.8 1.2
According to the linear theory, the wavelength of the lee-wave can be approximated by
(20)
= 21rU/N.
Though three-dimensional waves have complicated structures, the wavelength in the flow direction was the same as that in the two-dimensional case (HS; Castro et al., 1983). To compare the computed results with eq. (20), the wavelength of the lee-wave is determined from the first wave of surface streamlines because the wave amplitude decays rapidly. We tabulate it in table 1. The present results are in good agreement with those obtained from the linear theory. HS investigated the relation between the lee-waves and the separation. When the lee-wavelength X is much larger than hill length L (not height), the separation is boundary-controlled. The separation is suppressed when ?, is of the same order as L. As X becomes shorter than L, separation is induced by the lee-wave and occurs just upstream of the first lee-wave trough. HS
(a)
...-~__-=-~:S£~:(-
",.2 ~ _ - .
.
.
.
.
.
-~.--~=.=~.
.......... :~_~
(b)
_ .-~=----=--- / . . . . '. ~Y-', X'\ -- : . - ~ 1 I ,~.~.-,-y~.~\ '
- :~
I .:.--~
~aT~;,.!:~'-'~ ""k.i~,~ ',~ ..
, ";
_--4"--:>2-:~~\ - ~ .... -..<;-----~_ ~..~ L-_---~ . . . . . . "--=:-- ;, - > L . -v~'s~-'~"-:~--~ -~
%.,~.'~--~-~.=-
..~-
~_-___-=~-~.--
(-._-
_ ~
- - _ ~
- - ~
..... "-- . . . . . . . .
5-:' ~ - ~ -=-~--~ -~--~__
-
.
.
.
.
:--
.
.
(c)
Fig. 5. Contours of [ ~01 in the centre-plane ( y = 0 p l a n e ) ( R e = 10 ~, t = 30). (a) F = 1.0, (b) F = 0.5, (c) F = 0.2. The values are 2 n ( n = 0, 1, 2).
11
M. Suzuki, K. Kuwahara / Strati['ied ['low past a bell-shaped hill
Approaching flow
/
......... . ~ '
/
(a)
Approaching flow ...., )
_
--
e
....¢~.
~
.
/;, ( ~ . . . 9
~-.-w¢~
",
~\
"
"
(b)
_/.-:.
Approaching flow
i IY
/
~ " ~:% 2.<~_~;-7~.-~-~~ .'
.........."----
~
:i; +
~"
/"
~--f"-
(c)
Fig. 6. Contours of I~0l in the cross sections at x = O , 2h, 4h, 68, 8h and 10h ( R e = 1 0 ~, t = 3 0 ) . (a) F = I . 0 , (b) F = 0.5, (c) F = 0.2. The values are 2 = (n = 0, 1, 2). The view is from the upper left at the back of the hill. The origin of the axes (x, y, z) is located at the centre of the hill.
12
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
A p p r o a c h i n g flow
/;?( ~ /
,2
/
5
/
(,~g
/
2:? • "
Fig. 7. Contours of the x-component (the flow direction) of vorticity, ]~ ] in the cross sections at x = 0, 0.5h, 1.5h, 2.5h, 3.5h, 4.5h and 5.5h ( F = 0.5, Re =10 s, t = 30). The values are _+2n (n = 0, 1, 2). The view is from the upper left at the back of the hill. The origin of the axes (x, y, z) is located at the centre of the hill.
observed that the critical Froude number at which separation is suppressed, is around F = 1.0. This agrees with our result and the separation points in the centre-plane ( y = 0 plane) also confirm the above discussion. Figure 5 presents contours of I~0l, the absolute magnitude of vorticity, in the centre-plane ( y = 0 plane) at F = 1.0, 0.5 and 0.2. Figure 6 also shows contours of I~01 in the cross sections (x = 0, 2h, 4h, 6h, 8h, 10h) at F - - 1.0, 0.5 and 0.2. The view is from the upper left at the back of the hill and flow comes from the upper left of the pictures. There are strong vorticities under the lee-wave crest and near the lower boundary. The cross-streamwise structure of vorticity shows that its width is almost the same as the horizontal length of the hill. This is in reasonable agreement with the prediction of the three-dimensional linear theory by Crapper (1959), the experiments of Castro (1987) and the computation of Hanazaki (1988). In the case of F = 0.2 (fig. 5c), strong vorticity exists in two regions of the wake: near the lower boundary and at the dividing-streamline. The former indicates the boundary layer. The latter denotes there is strong shear between the free-stream and the horizontal recirculating flow, and the resulting vertical vortices. Considering the surface streamlines in the centre-plane (fig. 4c), this agrees with Drazin's theory: the flow is not restricted to move horizontally in the region T. The thickness of the region T, which is determined from the width of the vertical vortices, is 0.2h-0.25h by rough estimate. This value agrees with Drazin's theory, which predicts 0.2h. Figure 7 is a close up view of the contours of the x-component (the flow direction) of vorticity, 0~, in the cross sections (x = 0, 0.5h, 1.5h, 2.5h, 3.5h, 4.5h, 5.5h) at F = 0.5. Two strong, nearly stationary ¢0x regions exist in the cross section at x = 3.5h (arrows in fig. 7). Considering the reverse flows under the first lee-wave crest in the centre-plane (fig. 4b), it may have some relevance to the cowhorn-eddy which was observed by Brighton (1978).
3.3. The horizontal recirculating flow As the Froude number becomes smaller, the region H~, in which the fluid moves approximately in the horizontal direction, spreads (fig. 4c). The top view of surface streamlines in the grid surface of ~"= 12, 20 and 32 are presented in fig. 8. Each grid surface of ~"= constant on
M. Suzuki, K. Kuwahara / S t r a t i f i e d f l o w p a s t a bell-shaped hill
13
fC=32 = 20
~..
_--
- . . . . "-~..,.. " - , ' < ~
\
,
C = 12
-
f
~
I (b)
5
..............
i(c)
Id) Fig. 8. Top view of surface streamlines in grid surfaces of ~ = constant ( F = 0.2, Re = 103, t = 30). (a) Grid system (side view), (b) ~"= 12 grid surface, (c) ~"= 20 grid surface, (d) ~"= 32 grid surface. In ( b - d ) , the surface heights at some distance from the hill ( x > 3h) are approximately 0.05h, 0.2h and 0.8h, respectively. The origin of axes (x, y ) is located at the centre of the hill in the top view.
the centre-plane is shown in fig. 8a. The surface heights at a distance from the hill (x > 3h) are approximately 0.05h, 0.2h and 0.8h, respectively. These shows the three-dimensional wake structure. Above the hill surface a pair of vortex separation points (Lighthill, 1963) are
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
14
observed (fig. 8b). They form twin vortex (fig. 8c). The length of this recirculating region in the flow direction is much longer than that of the wake behind the two-dimensional circular cylinder in the neutral flow. The twin vortex is diffused at the dividing-streamline height (fig. 8d). Though we conducted time integration until t = 200, this twin vortex remains almost steady. In the case of F = 0.1, the twin vortex becomes unsteady and then vortices start to shed almost periodically after t = 50. ~ top view of time development of surface streamlines in the I
20 grid surface
=
I
/
l (a)
(b)
"~-_
"~
~ ".,
-'---.
r~'..'7;
~ - -
~
," ,) . I! i /!t
--
~
~
~
~
~S~//
~
,il//
/.~-/~.~v
. . . . . .
i ~
%%~.
Ldi I
A
\
~
I
~.\ "q
,/" f t
"J (C)
_
i -
~
-
~
-
~
_
~
/,s ,. "///~f.,<
Vs ---
.......
~-;~-~----~,
x
\\ ".-~- - " - - - ' - ~ - ~ - ~ / J _ " - - ' V
-j
Fig. 9. T o p view of time development of surface streamlines in the grid surface of ~'= 20. ( F - 0 . 1 and R e = I 0 ~) (a) Grid system (side view), (h) t = 72, (c) t = 76, (d) t = 80. T h e surface height at s o m e distance from the hill (x > 3h) is appro×imately 0.2h. T h e origin of the a×es (x, y) is located at the centre of the hill in the top view.
M. Suzuki, K. Kuwahara / S t r a t i f i e d f l o w past a bell-shaped hill
15
grid surface of ~ = 20 (z = 0.2h at a distance f r o m the hill, x > 3h) at F = 0.1 is shown in fig. 9. These pictures show almost a one-half cycle of vortex shedding. HS reported the oscillation of the flow in the wake at F = 0.1. According to Brighton (1978), the vortex shedding occurs at F_< 0.15 irrespective of the obstacle shape. But Castro et al. (1983) observed it at F < 0.3 with a triangular-shaped ridge and stated that a larger critical F r o u d e n u m b e r than Brighton's m a y result from the larger Reynolds n u m b e r in their experiments. O u r c o m p u t a t i o n s show that vortex shedding occurs at F = 0.1 but not at F = 0.2, although they are c o n d u c t e d at Re = l0 s, a larger Reynolds n u m b e r than in any of the above experiments. Therefore the determination of the critical F r o u d e n u m b e r needs further investigations. T h o u g h the vortex shedding period is not constant during our time integration, the Strouhal n u m b e r L / U T based on the shedding period T, approaching velocity U and the local hill diameter L at z = 0.2h, i.e. L = 2 r ( z = 0.2h) = 2.8h, is about 0.2 for t = 72-120. Since the sampling time is short, discussions on this Strouhal n u m b e r should be suspended. This value is, however, almost the same as that of the two-dimensional circular cylinder for Re = 1 0 3 - 2 × 105. The structure of the wake is seen in the top view of surface streamlines in grid surfaces of ~"= 8, 16, 20, 26 and 32 at F = 0.1 in fig. 10. The surface heights at a distance from the hill (x > 0.3h) are approximately 0.02h, 0.1h, 0.2h, 0.4h, and 0.8h, respectively. T h o u g h there are singular points which indicate vortex separation on the hill surface (fig. 10b), they do not f o r m a symmetric pair as seen in the case of F = 0.2 (fig. 8b). The vortex shedding at different levels in H 1 is o b t a i n e d in figs. 10c-10e. ¢ = 26
C = 32
(a)
7 = 8
%
.....
\
~
\, \
..... -- ..._
k-.
..
",
¢ = 16
x'~.
-__ \
t,/
I.7,:
......~ . ~ _ ~
. . . .
C = 20
--J.S>
\
,/
~
~ -
d(b)
c) Fig. 10. Top view of surface streamlines in grid surfaces of ~"= c o n s t a n t ( F = 0.1, Re = 103, t = 72). (a) Grid system (side view), (b) ~"= 8 grid surface, (c) ~"= 16 grid surface, (d) ~ = 20 grid surface, (e) ~"= 26 grid surface, (f) ~"= 32 grid surface. In (b-f), the surface heights at s o m e distance from the hill ( x > 3h) are a p p r o x i m a t e l y 0.02h, 0.1h, 0.2h, 0.4h and 0.8h, respectively. The origin of axes (x, y ) is located at the centre of the hill in the top view.
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
16
---~-'<',~
------~
.
i
~
~
-
,,r.,,
--.._
_ ~
"
--~
~
~,.,~.,
.,...
.
%
\%\.\N d)
i ~-1, J--
. __ .
.
.
~
~'*
~ - ~ , , , ~~ %
/I/TL"..------~. I
" k/# l / /7 / ~ < . . - ~ - -~'~, :~ 7 1/ .l/l!Ag
~..~%.~.~-..I
le)
Fig. 10 (continued).
The vortex indicated by the arrows moves downstream at higher levels. This is supposed to be the phase variation in vortex shedding with height, which Brighton (1978) mentioned in his paper. Brighton (1978) also stated that the shedding frequency is the same at all heights, whereas Castro et al. (1983) suggested that the vortex shedding frequency varies with height. They observed that the convection speed Uc of the vortices at z = 0.5h is twice as much as U c at z = h. They partly attributed its mechanism to upstream colunmar disturbance. Our computation with the infinite upper boundary, which is expected to have no columnar disturbance, shows no notable variance of the vortex shedding frequency with elevation during our observational period.
4. Conclusions The stratified flow past a bell-shaped hill has been studied numerically. The general pl]enomena with different Froude numbers have been examined. The results are in good qualitative as well as quantitative agreement with the theoretical and experimental results. The
M. SuzukL K. Kuwahara / Stratified flow past a bell-shaped hill
17
three-dimensional wake structure is successfully obtained. In particular, we have shown the following. (1) HS' formula HD = h(1 - F ) for F < 1.0 is confirmed. (2) Separation is suppressed at around F = 1.0, while separation is induced by the lee-wave and occurs just upstream of the first lee-wave trough at F < 1.0, as was shown by HS. The wavelength of the lee-wave is approximated by the linear theory. The width of the lee-wave in the cross-streamwise direction is almost the same as the horizontal length of the hill. (3) At low Froude numbers the flow is mainly horizontal, except in the regions near the lower boundary and the dividing-streamline, which have thickness of about Fh, as described by Drazin's theory. In the case of F = 0.2, a twin vortex exists in the horizontal recirculating region. It originates from a pair of vortex separation points above the hill surface and diffuses at the dividing-streamline height. While this twin vortex remains almost steady, vortex shedding occurs at F = 0.1. The Strouhal number is about 0.2. Phase variation in vortex shedding with height is observed, while no variance of shedding frequency with height is found. In this study stably linear density gradients and uniform approaching flow are considered. In the atmosphere, however, this situation seldom occurs. The local heating and cooling of the hill surface is another important factor whose effect has rarely been investigated. The present study may serve as a guide for understanding the real situation.
Acknowledgement The authors would like to thank Dr. K. Ishii of the Institute of Computational Fluid Dynamics (iCFD) for his advice. They are also grateful to Dr. S. Shirayama of iCFD. The computer program has been developed from his code.
References Baines, P.G. (1987) Upstream blocking and airflow over mountains, Ann. Rev. Fluid Mech. 19, 75-97. Berger, E. and R. Wille (1972) Periodic flow phenomena, Ann. Rev. Fluid Mech. 2, 313-338. Brighton, P.W.M. (1978) Strongly stratified flow past three-dimensional obstacles, Q. J. Roy. Met. Soe. 104 289-307. Castro, I.P. (1987) A note on lee wave structures in stratified flow over three-dimensional obstacles, Tellus 39A, 72-81. Castro, I.P., W.H. Snyder and G.L. Marsh (1983) Stratified flow over three-dimensional ridges, J. Fluid Mech. 135, 261-282. Crapper, G.D. (1959) A three-dimensional solution for waves in the lee of mountains, J. Fluid Mech. 6, 51-76, Drazin, P.G. (1961) On the steady flow of a fluid of variable density past an obstacle, Tellus 8, 239-251. Hanazaki, H. (1988) A numerical study of three-dimensional stratified flow past a sphere, J. Fluid Mech. 192, 393-419. Harlow, F.H. and J.E. Welch (1965) Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8, 2182-2189. Hinze, J.O. (1975) Turbulence (McGraw-Hill, New York). Hunt, J.C.R., D.P. Lalas and D.N. Asimakopoulos (1984) Air flow and dispersion in rough terrain: a report on Euromech 173, J. Fluid Mech. 142, 201-216. Hunt, J.C.R. and W.H. Snyder (1980) Experiments on stably and neutrally stratified flow over a model three-dimensional hill, J. Fluid Mech. 96, 671-704. Janowitz, G.S. (1981) Stratified flow over a bounded obstacle in a channel of finite height, J. Fluid Mech. 110, 161-170. Kawamura, K., H. Takami and K. Kuwahara (1986) Computation of high Reyriolds number flow around a circular cylinder with surface roughness, Fluid Dyn. Res. 1, 145-162. Lighthill, M.J. (1963) Laminar Boundary Layers, ed. L. Rosenhead (Oxford Univ. Press, Oxford). Long, R,R. (1957) Some aspects of the flow of stratified fluids: I. A theoretical investigation, Tellus 5, 42-57. Lugt, H.J. (1983) Vortex Flow in Nature and Technology (Wiley, New York). Nakamura, S. and M. Suzuki (1987) Noniterative three-dimensional grid generation using a parabolic-hyperbolic scheme, AIAA paper 87-0277.
18
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
Obayashi, S. and K. Kuwahara (1986) Numerical methods for the compressible Navier-Stokes equations, Nagare 5, 130-146 [in Japanese]. Queney, P. (1948) The problem of airflow over mountains: a summary of theoretical studies, Bull. Am. Met. Soc. 29, 16. Roache, P.J. (1989) The need for control of numerical accuracy, AIAA paper 89-1669. Ruscher, P.H. and J.W. Deardorff (1982) A numerical simulation of an atmospheric vortex street, Tellus 34, 555-566. Sheppard, P.A. (1956) Airflow over mountains, Q. J. Roy. Met. Soc. 82, 528-529. Smolarkiewicz, P.K. and R. Rotunno (|989) Low Froude number flow past three-dimensional obstacles. Part 1: Baroclinically generated lee vortices, J~ )limos. Sci. 46, 1154-1164. Snyder, W.H. (1981) Guideline for fluid modeling of atmospheric diffusion, in: Rep. No. EPA-600/8-81-O09 (Environ. Prot. Agency, Research Triangle Park NC). Snyder, W.H. (1985) Fluid modeling of pollutant transport and diffusion in stably stratified flows over complex terrain, Ann. Rev. Fluid Mech. 17, 239-266. Snyder, W.H., R.S. Thompson, R.E. Eskridge, R.E.Lawson, I.P. Castro, J.T. Lee, J.C.R. Hunt and Y. Ogawa (1985) The structure of strongly stratified flow over hills: dividing-streamlines concept, J. Fluid Mech. 152, 249-288. Sykes, R.I. (1978) Stratification effects in boundary layer flow over hills Proc. Roy. Soc. A 361,225-243. Thompson, J.F., Z.U.A. Warsi and C.W. Mastin (1985) Numerical Grid Generation (North-Holland, Amsterdam). Tsuchiya, K. (1969) The clouds with the shape of K~rmhn vortex street in the wake of Cheju island, Korea, J. Met. Soc. Japan 47, 457-465. Umeki, M. and Kambe, T. (1989) Stream patterns of an isothermal atmosphere over an isolated mountain, Fluid Dyn. Res. 5, 91-109.
FLU I D D ? N A M I C S RESEARCH
Stratified flow past a bell-shaped hill Masahiro Suzuki Institute of Computational Fluid Dynamics, 1-16-5 Haramachi, Meguro-ku, Tokyo 152, Japan
and Kunio Kuwahara The Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara-shi, Kanagawa 229, Japan
Received 9 March 1990 Abstract. A numerical study of the stratified flow past a bell-shaped hill is presented. The three-dimensional unsteady Navier-Stokes equations under the Boussinesq approximation are solved by a finite difference method in a generalized coordinate system. The characteristic phenomena of the stratified flow with various Froude numbers are successfullysimulated. The results are compared with theoretical and experimental data and good agreement is obtained.
1. Introduction
Stratified flow around surface-mounted hills is a subject that has attracted great interest from not only fluid dynamicists but also engineers in m a n y fields. It has strong relationships with industrial and environmental problems, as follows: (a) Air pollution problems, such as pollutant transport and dispersion over complex terrain (Snyder, 1985). (b) The prediction of the wake flow over mountains for aviation safety. The rotors behind mountains caused serious air accidents in the past. (c) The study of air flow over a complex relief necessary for effective wind energy utilization (Hunt et al., 1984). (d) Meteorological problems, for example, von Kfirmfin vortex street in the wake of an isolated island (Tsuchiya, 1969). An enormous number of investigations have been mainly conducted by the following four methods: (1) field study, (2) theoretical analysis, (3) experimental study, and (4) numerical study. Field study has contributed to discovery and understanding of the p h e n o m e n a of stratified flow since the 1920s, when glider pilots in the Alps discovered lee-waves (Lugt, 1983). In the 1960s the von Kfirm~n street-like vortex, which trails behind the islands in the earth's atmosphere, was discovered by a new tool of field study, i.e. meteorological satellites (Berger and Wille, 1972). ' Though many theoretical studies for two-dimensional flows have been conducted (Queney, 1948; Long, 1957), only a small number of results have been obtained for three-dimensional flows. For three-dimensional flows, linear perturbation analyses have been studied (Crapper, 1959; Umeki and Kambe, 1989), but nonlinear theoretical results, which can be applied to a 0169-5983/92/$5.50 © 1992 - The Japan Society of Fluid Mechanics. All rights reserved
2
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
.
.
.
.
.
~Fh
Fig. l. Small-Froude-number theory for gcr~,~;)vera three-dimensional hill by Drazin: Definition of regions (from Hunt and Snyder, 1980).
three-dimensional hill that is not a small perturbation of a plane surface, was obtained only by Drazin (1961). Drazin assumed as a first approximation that the flow is confined to horizontal planes and then developed a solution as a series of the small Froude number F. Brighton (1978) extended the solution to a higher order of the Froude number. For strongly stratified flow past obstacles, Sheppard (1956) proposed the formula
pu2( z = U p ) = g f o( h - z ) ( - d p / d z
) dz,
(1)
which is derived from the relation between the kinetic energy of the parcel, whose far upstream at elevation is HD, and the potential energy gained by it when being lifted through the density gradient. In eq. (1), U(z) is the approaching velocity, p(z) is the density of the fluid, h is the height of the obstacle, and g is the gravitational acceleration. If the fluid parcel is originally at a higher level than H D , it goes over the top of the obstacles, otherwise it may go around the sides. H D is called the "dividing-streamline height", which is a very important quantity for the assessment of pollutant concentration around mountains. Experimental approaches have been successful in explaining some of general phenomena. Hunt and Snyder (1980) (hereafter referred to as HS) conducted towing-tank experiments for a three-dimensional bell-shaped hill in a linearly stratified environment with an effectively uniform approaching flow. They verified that Drazin's theory is applicable over the Froude number range of F < 0.4, i.e. (1) The fluid moves approximately in the horizontal direction in the two regions H 1 and H 2. (See fig. 1 for the nomenclature.) (2) The flow is not restricted to move horizontally in regions B and T, whose thicknesses are about Fh. (3) In the region Hi, there exists mainly horizontal recirculation flow. They also investigated the dividing-streamline height and suggested the simple formula Hr, = h(1 - F ) ,
(2)
for 0 < F < 1. This formula can be derived from Sheppard's formula under the condition of uniform velocity and linear stratification. Unlike Sheppard's original formula, this does not have to be solved iteratively for the unknown H D. Snyder et al. (1985) verified Sheppard's formula for a variety of shapes with different upwind density and velocity profiles. They concluded that the formula provides a good estimate for H D for symmetric hills and small upwind shear. Brighton (1978) carried out experiments for a hemisphere, a cone, and a truncated cylinder in a closed-circuit stratified water channel. He observed that vortices are shed in region H t for F < 0.15 irrespective of the body shape and the Reynolds number (Re > 500), and the shedding frequency is the same at all heights. His experiments also showed a cowhorn-shaped eddy, which has a horizontal axis in the lee at the level of the top of the obstacle. He attributed its generation mechanism to the intensification o f . t h e eddy due to vortex stretching. A strong vertical shear exists where fluid, which clears the top of the obstacle
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
3
and separates from its lee slopes, flows over the stagnant horizontal recirculating flow region. A "multiple bubble", which is generated by an interaction between the flow of region T and the separated wake of region H1, and the lee wave crest together act as an effective blockage. They cause vortex stretching and the resulting vortex intensification. The vorticity is swept downstream in the trailing arms of the cowhorn eddy and is soon suppressed. But Castro et al. (1983) did not observe the cowhorn eddies in their experimental studies. They reported that vortex shedding is possible even at F = 0.3. Owing to the recent considerable advance in supercomputers, numerical studies have become powerful approaches compared with experiments. However, few numerical computations of three-dimensional stratified flow have been conducted. Sykes (1978) computed laminar flow over two-dimensional hills only. Ruscher and Deardorff (1982) applied a two-dimensional model of the atmospheric planetary boundary layer to simulating the development of a Kfirmfin vortex street downstream of the island of Cheju-do. Smolarkiewicz and Rotunno (1989) investigated the flow of a density-stratified fluid past a three-dimensional obstacle, using an inviscid, anelastic, nonhydrostatic finite difference model. Hanazaki (1988) conducted three-dimensional computation of viscous incompressible non-diffusive stratified flow past a sphere at Re = 200. But there have been no numerical studies on three-dimensional flow of higher Reynolds number. In this study we simulate stratified flow past a bell-shaped hill at high Reynolds numbers using the three-dimensional unsteady Navier-Stokes equations under the Boussinesq approximation. The governing equations are solved by a finite difference method using a generalized coordinate system. The results are compared with theoretical and experimental data. Our computations are conducted under the conditions considered by HS, which serve as reference data, and the three-dimensional structure of the wake is extensively investigated. 2. Numerical model and procedures
2.1. Model We consider a flow with vertical temperature gradients passing a bell-shaped obstacle on a fiat plate. The bell shape of the obstacle is defined as
f ( r ) = 1 / ( 1 + r4),
(3)
where f(r) is the height and r is the distance from the center of the obstacle. This shape is slightly different from the model used by HS. The approaching flow has stable and linear density gradients. No turbulence models are employed. In this study, the effect of the rotation of the earth is ignored, because the Rossby number is sufficiently large if the base diameter of the hill is less than approximately 5 km (Snyder, 1981).
2.2. Governing equations Under the Boussinesq approximation the dimensional governing equations are written as follows: Continuity equation: (4)
div' u' = 0. Momentum equation: 0u'
~~t
+ (u"V')u'
1 ~, g po(Z = O) grad' + po(Z = O) V 1
po(z = O) ( P ' - Po)gZ'.
p2
u
t
(5)
4
M. Suzukg K. Kuwahara / Stratified flow past a bell-shaped hill
Energy equation: 0T'
~.
Ot-----7-+(u"V')T'
,2T,
O,o(z=O)CpV
.
(6)
Equation of state for the incompressible fluid:
p'= p'o(Z = 01[1 - a(T~;C To')]. ¢
t
(7) t
¢
t
p
Here, u' is the velocity (u;, u2, u3) [ = (u , v , w')], t' is the time, O is the density, po(Z) is the given density distribution of the approaching flow, T ' is the temperature, To'(Z ) is the temperature distribution corresponding to p'o(Z), ~a, is the perturbation pressure, # is the viscosity, g is the acceleration due to gravity, z' is the unit vector along z-axis, )t is the thermal conductivity, a is an expansion coefficient, Cp is the specific heat at constant pressure, and the primes indicate dimensional values. We non-dimensionalize velocities by the uniform approaching velocity U, distances by the height of a hill h, time by h/U, density by Oo(Z --- 0), perturbation pressure by po(Z = 0 ) U 2 and temperature by To'(Z = 0). Using eq. (7) and assuming a = 1/To'(Z = 0) (perfect gas), the following non-dimensional equations are obtained: Continuity equation: div u = O.
(8)
Momentum equation:
Ou
1
0-t- + ( u ' V ) u =
- g r a d ~ + -R-TV
2R__
1 ( r ° - T)z
(9)
~
Energy equation:
OT 0-7 + ( u ' V ) T =
1 Re. Pr v2T"
(10)
Here, Re is the Reynolds number [ = Uhpo(Z = 0)//z], F is the Froude number ( = U/Nh), N is the Brunt-Vaisal~i frequency, and Pr is the Prandtl number ( = Ce~/Oo(Z = 0)~). The values of the air under the standard atmosphere at 300 K are used for p0(0), /~, A and Cp.
2.3. Computational method The numerical scheme is based on the well-known MAC method (Harlow and Welch, 1965). First, the Poisson equation for the pressure, derived by taking the divergence of eq. (9), is solved at time step n:
v 2 ~ "= -div[(u"'V)u"] -
div[ ( T ° - Tn)z] + A t '
(11)
where D = div u, superscript n denotes the non-dimensional time t = nat and At is the time increment of finite difference approximation of 3/Ot. Though D in eq. (11) is identically zero due to eq. (8), it is retained here to prevent the accumulation of numerical errors and to ensure the continuity equation implicitly. By the substitution of eq. (8) into the first term at the right-hand side, eq. (11) is rewritten in the form 3
3 3u~' 3u~
V2ga"= - Z
E Oxj Oxi
i=1
j=l
1
Dn
F~div[(T 0 - r")z] + A-7'
where (x l, x 2, x3) [ = (x, y, z)] are the variables in the physical plane.
(12)
M. Suzuki, K. Kuwahara /Stratifiedflow past a bell-shaped hill
5
Secondly, u and T at the next time step are obtained by solving the Navier-Stokes and the energy equations, un+l
At
Un
+ ( u n ' V ) u " + 1 = - g r a d ~ n + ~_~ V2un+l_ ~F1 (To- T")z,
T n+l - T" At +(u"'V)Tn+I-
1 - - Pr V ZT'+l Re.
(13) (14)
The computations are done on a generalized coordinate system, which is obtained from the following coordinate transformations (Thompson, et al., 1985):
xa=x,(~', ~z, ~3),
x 2 = x 2 ( ~ , ' ~2, ~3),
x3=x3(~a, ~2, ~3),
(15)
where (~1, ~2, ~3) [= (~, ~/, ~)] are the coordinates in the transformed plane. Equations (12)-(14) are written in the generalized coordinates as
E ~ =-
g,j
+ .i
E E
/=1 j=l
1
ai"
-~
a J" ~
-~
-
i=lE ai"
(TO- T")Z]
3
+ ~ Y'. a i" Ou----~ i=1
(16)
O,f '
3 aiO(uk)n+l (."+l)k -- (.")k At + i=1 y'~ u"" a~ ~
= - E (i=1 a)k-~7-+-~E 1
Fz(To-T")(z)k T n+l - T"
At
- R
3
g'J O~ i ~)~Y.+ a'
0~' ]
(k=1,2,3),
(un+l)k
(17)
0T,+1
+ E un " ai i=l
Pr i=1 y'~ =
i=1 j E =l
~i
gij a~ i a ~ . + a i, a~' ! -gC rn + ~'
(18)
where a ~ ( = V~ ~) is the contravariant base vector, g~J ( = a ~, a j) is the contravariant metric tensor, and subscript k denotes the component in the k-direction. The base vectors are represented by the second-order central-differences scheme. All spatial derivatives, except nonlinear terms in (16)-(18), are approximated by the second-order central-differences scheme. The nonlinear terms are represented by a third-order upwind scheme (Kawamura, et al., 1986), .i.e.
(f~_~3u)i-Ji-fui+2-2ui+l+9ui-lOui6 A~i
1+2ui
2
=f~-2ui+2+lOu'+l-9U~+2u'-l-ui-2
for ~ > 0, for f / < 0 ,
(19)
6 A~i
where f is an arbitrary function and subscript i denotes the grid point. These difference equations are solved by a modified SOR (successive over-relaxation) method which contains no recursive references in the DO loop and achieves high vectorization on supercomputers.
6
M. Suzuki, K. Kuwahara / S t r a t i f i e d f l o w past a bell-shaped hill
2.4. Grid system In figure 2 the grid system is presented. The number of grid points distributed to each coordinate direction is 173, 89 and 63 in the ~, , / a n d ~"directions, respectively. The surface grid system on the hill and floor (~" = ~'0) is first formed, then the grid system along ~"= constant is heaped up in the ~'-direction (in this case 62 times) by a parabolic-hyperbolic hybrid grid generation scheme (Nakamura and Suzuki, 1987). [These grid surfaces of ~"= constant are not necessarily parallel to the plane of • = constant]. The grid points must cluster near the lower boundary to calculate the boundary layer accurately. The minimum spacing between two neighboring grid points is customarily fixed at 0.1/RvrR-e (Obayashi and Kuwahara, 1986). Therefore, the minimum spacing is set to be approximately 3 x 10 -4. In order to ascertain grid convergence, computations on a finer grid are tested. The method of grid doubling in each direction was proposed to ascertain grid convergence (Roache, 1989). However, increasing the number of grid points in each direction by a factor of 2, i.e. a total increase by a factor of 8, is not feasible with our computational facilities. Therefore, at this stage, a grid of 215 x 119 x 80, which is the maximum grid point number with our computer system, is used. No significant differences in the large structures of the flow field between results on the finer grid and these on the original grid are found. Though the test does not prove grid convergence, it confirms that a grid of 173 x 89 X 63 is fine enough to obtain the general features of the flow. A more accurate test remains a task for future computers.
2.5. Initial and boundary conditions We adapt the following initial condition as the impulsive start: u = (1, 0, 0), ~a= 0 and T = To(z ) in all regions. The boundary conditions are as follows.
'r
A p p r o a c h i n g flow
~"---~1
2 h
(a) 1Lli] ~ ~ ] i i t
111t i li~lll rlli i i ]
L+[~
~ ',,.;,,w,', ',4,,M~
zI
@
•",
,
, <~
,
,
~
Ib) Fig. 2. Grid system (173 x 89x63 grid points). (a) Hill surface and side, outflow boundaries, (b) side view of the centre-plane (y = 0 plane). [Note: the origin of the physical coordinate system (x, y, z) is located at the centre of the
hill].
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
7
On the outer boundary except upstream: u, ~ and T are determined by an extrapolation. On the upstream boundary: u = (1, 0, 0), ~ = 0 and T = To(z ). On the lower boundary: u = (0, 0, 0), while ~ and T are determined by the extrapolation. The outer boundary condition ensures the condition of infinite boundaries. According to the linear theory (Janowitz, 1981), the columnar disturbance becomes weaker as h / D ( D is the half-depth of the channel) becomes small and, finally, is expected to disappear at D ~ oo (infinite upper boundary). Though the upstream influence is observed in the stratified flow (Baines, 1987), the upstream condition is fixed to be uniform. Because of the remoteness of the upstream boundary from the hill and also because of the rough mesh in the upstream region, the upstream influence will be damped before reaching the upstream boundary. To identify the flow variations with Froude numbers, the cases of F = co, 1.5, 1.0, 0.5, 0.2, and 0.1 are calculated at Re = 105. Computations are mainly done on the Fujitsu VP-400E, which is a 1700 Mflops supercomputer at the Institute of Computational Fluid Dynamics. Typical CPU time is about 10-20 h for each case.
3. Results and discussion
We mostly observed quasi-stationary flow patterns in stably stratified flow after non-dimensional time t --- 25 except for vortex shedding in the horizontal recirculating region. Thus, some of the typical instantaneous flow patterns are presented here. 3.1. The dividing-streamline height
Computed particle traces emerging from upstream at several levels in the centre-plane ( y = 0 plane) are obtained to examine dividing-streamline heights for 0 < F < 1. The results are plotted in fig. 3. Because the approaching flow has a uniform velocity at the upstream boundary and the no-slip condition is applied on the lower boundary, the boundary layer may develop depending on the Reynolds and the Froude numbers as the flow reaches the hill. In the absence of a hill, i.e. a flat plate, the flow in the boundary layer is characterized by the Reynolds number Re~ with reference to the distance from the upstream edge of the flat plate. Since the distance from the upstream boundary to the centre of the hill is 12h and the Reynolds number Re based on the hill height is 105 , Re, = U12hpo(Z = 0)//z = 12 Re = 1.2
x
106.
This is less than the critical value of Re~ for the onset of turbulence in the laminar boundary
1.0
0.5
~, \\ \\
i \\
o.
o.5
"\\
\\
\
1.0 F Fig. 3. Dividing-streamlineheight versus Froude number (Re ~ 105). The drawn line shows HD/h = 1 F.
8
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
layer on a flat plate (Hinze, 1975). Thus, the flow in the boundary layer can be expected to be laminar at the centre of the hill (but in the absence of it). According to Blasius' solution for the plane laminar boundary layer, the boundary layer thickness 8, at which the velocity equals 99.7% of the constant free-stream velocity, can be obtained: 8 = 5.5 x
12h/R~x= 0.06h. +++.~+
i
- - ~ ~ + +
-'+~-~~'---~-~--
....
~ +
Fig. 4. Surface streamlines in the centre-plane ( y = 0 plane) a n d surface shear-stress p a t t e r n s o n the lower b o u n d a r y (Re = 105, t = 30), (a) F = 1.0, (b) F = 0.5, (c) F = 0.2. The origin of the axes (x, y ) is located ai the centre of the hill in the surface shear-stress pattern.
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
Fig. 4 (continued).
The computations, which are done on a grid with the same distribution of points but in the absence of the hill, show approximately the same degree of boundary layer thickness. No significant variations of the boundary layer thickness with the Froude numbers are found. Therefore the boundary layer is thin enough, and the present results are in good agreement with HS' formula H D = h(1 - F), which is valid for effectively uniformity approaching flows. Slight deviations at a higher Froude numbers ( F = 0.5 and 1.0) are mainly due to the existence of horseshoe vortices (arrows in figs. 4a and 4b), which produce some downwash on the upwind slope of the hill (Castro et al., 1983). 3.2. The lee-wave structure
The surface streamlines in the centre-plane ( y = 0 plane) and the surface shear-stress patterns on the lower boundary at F = 1.0, 0.5 and 0.2 are shown in fig. 4. At a Froude number of 1.0, the fluid flows along the lee slope without separating. Separation begins farther downstream of the hill. At a Froude number of 0.5, separation begins roughly half-way down the lee slope. There are reverse flows under the lee-wave. The amplitude of the lee-wave decreases rapidly in the flow direction. Some of the streamlines do not go over the top of the hill. At a Froude number of 0.2, the flow separates just when it passes the top of the hill. The range of heights at which the streamlines go over the top of the hill shrinks. The vertical movement of the flow is suppressed. Though the locations of separation lines, which are observed in the surface shear-stress patterns, coincide with HS' results, the surface shear-stress patterns in the wake are different from those of the HS. The patterns are more complex. No symmetric pair of prominent singular points is clearly found in the wake. There are many small singular points. Since HS drew the surface shear-stress patterns by collating all observations, the patterns are somewhat speculative mean patterns. Accordingly, the differences in the surface shear-stress patterns between HS and the present computations are mainly due to the difference between instantaneous and time averaged flow patterns.
M. Suzuki, K. Kuwahara
10
/Stratifiedflowpast
a bell-shaped hill
Table 1 Comparison of theoretical and observed lee-wavelength ?~ Froude number
X/h
theory 1.0 0.5 0.2
observed
6.28 3.14 1.26 ~,~,
6.0 2.8 1.2
According to the linear theory, the wavelength of the lee-wave can be approximated by
(20)
= 21rU/N.
Though three-dimensional waves have complicated structures, the wavelength in the flow direction was the same as that in the two-dimensional case (HS; Castro et al., 1983). To compare the computed results with eq. (20), the wavelength of the lee-wave is determined from the first wave of surface streamlines because the wave amplitude decays rapidly. We tabulate it in table 1. The present results are in good agreement with those obtained from the linear theory. HS investigated the relation between the lee-waves and the separation. When the lee-wavelength X is much larger than hill length L (not height), the separation is boundary-controlled. The separation is suppressed when ?, is of the same order as L. As X becomes shorter than L, separation is induced by the lee-wave and occurs just upstream of the first lee-wave trough. HS
(a)
...-~__-=-~:S£~:(-
",.2 ~ _ - .
.
.
.
.
.
-~.--~=.=~.
.......... :~_~
(b)
_ .-~=----=--- / . . . . '. ~Y-', X'\ -- : . - ~ 1 I ,~.~.-,-y~.~\ '
- :~
I .:.--~
~aT~;,.!:~'-'~ ""k.i~,~ ',~ ..
, ";
_--4"--:>2-:~~\ - ~ .... -..<;-----~_ ~..~ L-_---~ . . . . . . "--=:-- ;, - > L . -v~'s~-'~"-:~--~ -~
%.,~.'~--~-~.=-
..~-
~_-___-=~-~.--
(-._-
_ ~
- - _ ~
- - ~
..... "-- . . . . . . . .
5-:' ~ - ~ -=-~--~ -~--~__
-
.
.
.
.
:--
.
.
(c)
Fig. 5. Contours of [ ~01 in the centre-plane ( y = 0 p l a n e ) ( R e = 10 ~, t = 30). (a) F = 1.0, (b) F = 0.5, (c) F = 0.2. The values are 2 n ( n = 0, 1, 2).
11
M. Suzuki, K. Kuwahara / Strati['ied ['low past a bell-shaped hill
Approaching flow
/
......... . ~ '
/
(a)
Approaching flow ...., )
_
--
e
....¢~.
~
.
/;, ( ~ . . . 9
~-.-w¢~
",
~\
"
"
(b)
_/.-:.
Approaching flow
i IY
/
~ " ~:% 2.<~_~;-7~.-~-~~ .'
.........."----
~
:i; +
~"
/"
~--f"-
(c)
Fig. 6. Contours of I~0l in the cross sections at x = O , 2h, 4h, 68, 8h and 10h ( R e = 1 0 ~, t = 3 0 ) . (a) F = I . 0 , (b) F = 0.5, (c) F = 0.2. The values are 2 = (n = 0, 1, 2). The view is from the upper left at the back of the hill. The origin of the axes (x, y, z) is located at the centre of the hill.
12
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
A p p r o a c h i n g flow
/;?( ~ /
,2
/
5
/
(,~g
/
2:? • "
Fig. 7. Contours of the x-component (the flow direction) of vorticity, ]~ ] in the cross sections at x = 0, 0.5h, 1.5h, 2.5h, 3.5h, 4.5h and 5.5h ( F = 0.5, Re =10 s, t = 30). The values are _+2n (n = 0, 1, 2). The view is from the upper left at the back of the hill. The origin of the axes (x, y, z) is located at the centre of the hill.
observed that the critical Froude number at which separation is suppressed, is around F = 1.0. This agrees with our result and the separation points in the centre-plane ( y = 0 plane) also confirm the above discussion. Figure 5 presents contours of I~0l, the absolute magnitude of vorticity, in the centre-plane ( y = 0 plane) at F = 1.0, 0.5 and 0.2. Figure 6 also shows contours of I~01 in the cross sections (x = 0, 2h, 4h, 6h, 8h, 10h) at F - - 1.0, 0.5 and 0.2. The view is from the upper left at the back of the hill and flow comes from the upper left of the pictures. There are strong vorticities under the lee-wave crest and near the lower boundary. The cross-streamwise structure of vorticity shows that its width is almost the same as the horizontal length of the hill. This is in reasonable agreement with the prediction of the three-dimensional linear theory by Crapper (1959), the experiments of Castro (1987) and the computation of Hanazaki (1988). In the case of F = 0.2 (fig. 5c), strong vorticity exists in two regions of the wake: near the lower boundary and at the dividing-streamline. The former indicates the boundary layer. The latter denotes there is strong shear between the free-stream and the horizontal recirculating flow, and the resulting vertical vortices. Considering the surface streamlines in the centre-plane (fig. 4c), this agrees with Drazin's theory: the flow is not restricted to move horizontally in the region T. The thickness of the region T, which is determined from the width of the vertical vortices, is 0.2h-0.25h by rough estimate. This value agrees with Drazin's theory, which predicts 0.2h. Figure 7 is a close up view of the contours of the x-component (the flow direction) of vorticity, 0~, in the cross sections (x = 0, 0.5h, 1.5h, 2.5h, 3.5h, 4.5h, 5.5h) at F = 0.5. Two strong, nearly stationary ¢0x regions exist in the cross section at x = 3.5h (arrows in fig. 7). Considering the reverse flows under the first lee-wave crest in the centre-plane (fig. 4b), it may have some relevance to the cowhorn-eddy which was observed by Brighton (1978).
3.3. The horizontal recirculating flow As the Froude number becomes smaller, the region H~, in which the fluid moves approximately in the horizontal direction, spreads (fig. 4c). The top view of surface streamlines in the grid surface of ~"= 12, 20 and 32 are presented in fig. 8. Each grid surface of ~"= constant on
M. Suzuki, K. Kuwahara / S t r a t i f i e d f l o w p a s t a bell-shaped hill
13
fC=32 = 20
~..
_--
- . . . . "-~..,.. " - , ' < ~
\
,
C = 12
-
f
~
I (b)
5
..............
i(c)
Id) Fig. 8. Top view of surface streamlines in grid surfaces of ~ = constant ( F = 0.2, Re = 103, t = 30). (a) Grid system (side view), (b) ~"= 12 grid surface, (c) ~"= 20 grid surface, (d) ~"= 32 grid surface. In ( b - d ) , the surface heights at some distance from the hill ( x > 3h) are approximately 0.05h, 0.2h and 0.8h, respectively. The origin of axes (x, y ) is located at the centre of the hill in the top view.
the centre-plane is shown in fig. 8a. The surface heights at a distance from the hill (x > 3h) are approximately 0.05h, 0.2h and 0.8h, respectively. These shows the three-dimensional wake structure. Above the hill surface a pair of vortex separation points (Lighthill, 1963) are
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
14
observed (fig. 8b). They form twin vortex (fig. 8c). The length of this recirculating region in the flow direction is much longer than that of the wake behind the two-dimensional circular cylinder in the neutral flow. The twin vortex is diffused at the dividing-streamline height (fig. 8d). Though we conducted time integration until t = 200, this twin vortex remains almost steady. In the case of F = 0.1, the twin vortex becomes unsteady and then vortices start to shed almost periodically after t = 50. ~ top view of time development of surface streamlines in the I
20 grid surface
=
I
/
l (a)
(b)
"~-_
"~
~ ".,
-'---.
r~'..'7;
~ - -
~
," ,) . I! i /!t
--
~
~
~
~
~S~//
~
,il//
/.~-/~.~v
. . . . . .
i ~
%%~.
Ldi I
A
\
~
I
~.\ "q
,/" f t
"J (C)
_
i -
~
-
~
-
~
_
~
/,s ,. "///~f.,<
Vs ---
.......
~-;~-~----~,
x
\\ ".-~- - " - - - ' - ~ - ~ - ~ / J _ " - - ' V
-j
Fig. 9. T o p view of time development of surface streamlines in the grid surface of ~'= 20. ( F - 0 . 1 and R e = I 0 ~) (a) Grid system (side view), (h) t = 72, (c) t = 76, (d) t = 80. T h e surface height at s o m e distance from the hill (x > 3h) is appro×imately 0.2h. T h e origin of the a×es (x, y) is located at the centre of the hill in the top view.
M. Suzuki, K. Kuwahara / S t r a t i f i e d f l o w past a bell-shaped hill
15
grid surface of ~ = 20 (z = 0.2h at a distance f r o m the hill, x > 3h) at F = 0.1 is shown in fig. 9. These pictures show almost a one-half cycle of vortex shedding. HS reported the oscillation of the flow in the wake at F = 0.1. According to Brighton (1978), the vortex shedding occurs at F_< 0.15 irrespective of the obstacle shape. But Castro et al. (1983) observed it at F < 0.3 with a triangular-shaped ridge and stated that a larger critical F r o u d e n u m b e r than Brighton's m a y result from the larger Reynolds n u m b e r in their experiments. O u r c o m p u t a t i o n s show that vortex shedding occurs at F = 0.1 but not at F = 0.2, although they are c o n d u c t e d at Re = l0 s, a larger Reynolds n u m b e r than in any of the above experiments. Therefore the determination of the critical F r o u d e n u m b e r needs further investigations. T h o u g h the vortex shedding period is not constant during our time integration, the Strouhal n u m b e r L / U T based on the shedding period T, approaching velocity U and the local hill diameter L at z = 0.2h, i.e. L = 2 r ( z = 0.2h) = 2.8h, is about 0.2 for t = 72-120. Since the sampling time is short, discussions on this Strouhal n u m b e r should be suspended. This value is, however, almost the same as that of the two-dimensional circular cylinder for Re = 1 0 3 - 2 × 105. The structure of the wake is seen in the top view of surface streamlines in grid surfaces of ~"= 8, 16, 20, 26 and 32 at F = 0.1 in fig. 10. The surface heights at a distance from the hill (x > 0.3h) are approximately 0.02h, 0.1h, 0.2h, 0.4h, and 0.8h, respectively. T h o u g h there are singular points which indicate vortex separation on the hill surface (fig. 10b), they do not f o r m a symmetric pair as seen in the case of F = 0.2 (fig. 8b). The vortex shedding at different levels in H 1 is o b t a i n e d in figs. 10c-10e. ¢ = 26
C = 32
(a)
7 = 8
%
.....
\
~
\, \
..... -- ..._
k-.
..
",
¢ = 16
x'~.
-__ \
t,/
I.7,:
......~ . ~ _ ~
. . . .
C = 20
--J.S>
\
,/
~
~ -
d(b)
c) Fig. 10. Top view of surface streamlines in grid surfaces of ~"= c o n s t a n t ( F = 0.1, Re = 103, t = 72). (a) Grid system (side view), (b) ~"= 8 grid surface, (c) ~"= 16 grid surface, (d) ~ = 20 grid surface, (e) ~"= 26 grid surface, (f) ~"= 32 grid surface. In (b-f), the surface heights at s o m e distance from the hill ( x > 3h) are a p p r o x i m a t e l y 0.02h, 0.1h, 0.2h, 0.4h and 0.8h, respectively. The origin of axes (x, y ) is located at the centre of the hill in the top view.
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
16
---~-'<',~
------~
.
i
~
~
-
,,r.,,
--.._
_ ~
"
--~
~
~,.,~.,
.,...
.
%
\%\.\N d)
i ~-1, J--
. __ .
.
.
~
~'*
~ - ~ , , , ~~ %
/I/TL"..------~. I
" k/# l / /7 / ~ < . . - ~ - -~'~, :~ 7 1/ .l/l!Ag
~..~%.~.~-..I
le)
Fig. 10 (continued).
The vortex indicated by the arrows moves downstream at higher levels. This is supposed to be the phase variation in vortex shedding with height, which Brighton (1978) mentioned in his paper. Brighton (1978) also stated that the shedding frequency is the same at all heights, whereas Castro et al. (1983) suggested that the vortex shedding frequency varies with height. They observed that the convection speed Uc of the vortices at z = 0.5h is twice as much as U c at z = h. They partly attributed its mechanism to upstream colunmar disturbance. Our computation with the infinite upper boundary, which is expected to have no columnar disturbance, shows no notable variance of the vortex shedding frequency with elevation during our observational period.
4. Conclusions The stratified flow past a bell-shaped hill has been studied numerically. The general pl]enomena with different Froude numbers have been examined. The results are in good qualitative as well as quantitative agreement with the theoretical and experimental results. The
M. SuzukL K. Kuwahara / Stratified flow past a bell-shaped hill
17
three-dimensional wake structure is successfully obtained. In particular, we have shown the following. (1) HS' formula HD = h(1 - F ) for F < 1.0 is confirmed. (2) Separation is suppressed at around F = 1.0, while separation is induced by the lee-wave and occurs just upstream of the first lee-wave trough at F < 1.0, as was shown by HS. The wavelength of the lee-wave is approximated by the linear theory. The width of the lee-wave in the cross-streamwise direction is almost the same as the horizontal length of the hill. (3) At low Froude numbers the flow is mainly horizontal, except in the regions near the lower boundary and the dividing-streamline, which have thickness of about Fh, as described by Drazin's theory. In the case of F = 0.2, a twin vortex exists in the horizontal recirculating region. It originates from a pair of vortex separation points above the hill surface and diffuses at the dividing-streamline height. While this twin vortex remains almost steady, vortex shedding occurs at F = 0.1. The Strouhal number is about 0.2. Phase variation in vortex shedding with height is observed, while no variance of shedding frequency with height is found. In this study stably linear density gradients and uniform approaching flow are considered. In the atmosphere, however, this situation seldom occurs. The local heating and cooling of the hill surface is another important factor whose effect has rarely been investigated. The present study may serve as a guide for understanding the real situation.
Acknowledgement The authors would like to thank Dr. K. Ishii of the Institute of Computational Fluid Dynamics (iCFD) for his advice. They are also grateful to Dr. S. Shirayama of iCFD. The computer program has been developed from his code.
References Baines, P.G. (1987) Upstream blocking and airflow over mountains, Ann. Rev. Fluid Mech. 19, 75-97. Berger, E. and R. Wille (1972) Periodic flow phenomena, Ann. Rev. Fluid Mech. 2, 313-338. Brighton, P.W.M. (1978) Strongly stratified flow past three-dimensional obstacles, Q. J. Roy. Met. Soe. 104 289-307. Castro, I.P. (1987) A note on lee wave structures in stratified flow over three-dimensional obstacles, Tellus 39A, 72-81. Castro, I.P., W.H. Snyder and G.L. Marsh (1983) Stratified flow over three-dimensional ridges, J. Fluid Mech. 135, 261-282. Crapper, G.D. (1959) A three-dimensional solution for waves in the lee of mountains, J. Fluid Mech. 6, 51-76, Drazin, P.G. (1961) On the steady flow of a fluid of variable density past an obstacle, Tellus 8, 239-251. Hanazaki, H. (1988) A numerical study of three-dimensional stratified flow past a sphere, J. Fluid Mech. 192, 393-419. Harlow, F.H. and J.E. Welch (1965) Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8, 2182-2189. Hinze, J.O. (1975) Turbulence (McGraw-Hill, New York). Hunt, J.C.R., D.P. Lalas and D.N. Asimakopoulos (1984) Air flow and dispersion in rough terrain: a report on Euromech 173, J. Fluid Mech. 142, 201-216. Hunt, J.C.R. and W.H. Snyder (1980) Experiments on stably and neutrally stratified flow over a model three-dimensional hill, J. Fluid Mech. 96, 671-704. Janowitz, G.S. (1981) Stratified flow over a bounded obstacle in a channel of finite height, J. Fluid Mech. 110, 161-170. Kawamura, K., H. Takami and K. Kuwahara (1986) Computation of high Reyriolds number flow around a circular cylinder with surface roughness, Fluid Dyn. Res. 1, 145-162. Lighthill, M.J. (1963) Laminar Boundary Layers, ed. L. Rosenhead (Oxford Univ. Press, Oxford). Long, R,R. (1957) Some aspects of the flow of stratified fluids: I. A theoretical investigation, Tellus 5, 42-57. Lugt, H.J. (1983) Vortex Flow in Nature and Technology (Wiley, New York). Nakamura, S. and M. Suzuki (1987) Noniterative three-dimensional grid generation using a parabolic-hyperbolic scheme, AIAA paper 87-0277.
18
M. Suzuki, K. Kuwahara / Stratified flow past a bell-shaped hill
Obayashi, S. and K. Kuwahara (1986) Numerical methods for the compressible Navier-Stokes equations, Nagare 5, 130-146 [in Japanese]. Queney, P. (1948) The problem of airflow over mountains: a summary of theoretical studies, Bull. Am. Met. Soc. 29, 16. Roache, P.J. (1989) The need for control of numerical accuracy, AIAA paper 89-1669. Ruscher, P.H. and J.W. Deardorff (1982) A numerical simulation of an atmospheric vortex street, Tellus 34, 555-566. Sheppard, P.A. (1956) Airflow over mountains, Q. J. Roy. Met. Soc. 82, 528-529. Smolarkiewicz, P.K. and R. Rotunno (|989) Low Froude number flow past three-dimensional obstacles. Part 1: Baroclinically generated lee vortices, J~ )limos. Sci. 46, 1154-1164. Snyder, W.H. (1981) Guideline for fluid modeling of atmospheric diffusion, in: Rep. No. EPA-600/8-81-O09 (Environ. Prot. Agency, Research Triangle Park NC). Snyder, W.H. (1985) Fluid modeling of pollutant transport and diffusion in stably stratified flows over complex terrain, Ann. Rev. Fluid Mech. 17, 239-266. Snyder, W.H., R.S. Thompson, R.E. Eskridge, R.E.Lawson, I.P. Castro, J.T. Lee, J.C.R. Hunt and Y. Ogawa (1985) The structure of strongly stratified flow over hills: dividing-streamlines concept, J. Fluid Mech. 152, 249-288. Sykes, R.I. (1978) Stratification effects in boundary layer flow over hills Proc. Roy. Soc. A 361,225-243. Thompson, J.F., Z.U.A. Warsi and C.W. Mastin (1985) Numerical Grid Generation (North-Holland, Amsterdam). Tsuchiya, K. (1969) The clouds with the shape of K~rmhn vortex street in the wake of Cheju island, Korea, J. Met. Soc. Japan 47, 457-465. Umeki, M. and Kambe, T. (1989) Stream patterns of an isothermal atmosphere over an isolated mountain, Fluid Dyn. Res. 5, 91-109.