Bottom velocity computation method based on depth integrated model
without shallow water assumption
Tatsuhiko Uchida* and Shoji Fukuoka**
Research and Development Initiative,
Chuo University, Japan*
E-mail: utida@tamacc.chuo-u.ac.jp**
E-mail: sfuku@tamacc.chuo-u.ac.jp

Abstract
In order to compute local scouring around structure together with the computation of largescale flow phenomena, we have developed the Bottom Velocity Computation (BVC) method,in which depth-averaged horizontal vorticity equation and water surface velocity equation aresolved with shallow water equations to evaluate bottom velocity field and vertical velocity
distribution. The previous BVC method by the authors has been based on the shallow water
assumption, but the validity of the shallow water assumption for local flow structure has notbeen clarified. This paper presents applicable ranges of 2D, 3D and quasi-3D numericalmodel by the ratio of representative water depth to representative horizontal scale. Then, thegeneral BVC method based on a depth-integrated model without the shallow water
assumption is developed. The method is applied to flow field around a cylinder on flat roughbed measured by Roulund et al.(2005).
1.

Introduction

Local scouring around structure is one of the most important issues in the hydraulicengineering field. There are many researches on the local scouring. Flow fields aroundstructures inducing local scouring are so complex that many empirical formulae are proposed
based on experimental results (e.g., Hoffmans & Verheij, 1997). Recently, we can simulate
complex turbulence flow and local scouring by 3D turbulence model (e.g., Roulund et al.,2005). On the other hand, local scours around river structures during flood are affected by thelarger scale phenomena of flood flows and bed variations, such as discharge hydrograph,planner shapes of rivers and sand bars. So, to provide countermeasure against local scour
around river structures, it is to solve simultaneously not only local 3D flow around structures
but also large scale phenomena of flood flows. However, applications of the 3D model are
still limited to small scale phenomena such as bed variation in experimental channels, becauseof large computational time, many memories, and a lot of computational task.
Advanced depth integrated model, improved 2D model, is one of the effective method tocompute local scouring around structures with large-scale flow phenomena. Advanced depth
integrated models to compute bed variation include secondary flow model (e.g. Nishimoto etal., 1992) and a quasi-3D model (e.g. Fukuoka et al., 1992), in which vertical velocity
distributions are computed. Recently, we have developed a new quasi-3D method, the Bottom
Velocity Computation (BVC) method, in which depth-averaged horizontal vorticity and water1

surface velocity equations are computed with shallow water equations to evaluate bottom
velocity field and vertical velocity distribution (Uchida & Fukuoka, 2009, 2010, 2011). The
previous BVC method has been based on the shallow water assumption, in which
representative water depth h0 is much smaller than representative horizontal scale of theobjective phenomena L0 (shallowness parameter: s=h0/L0<<1). However, the validity of theassumption has not been clarified for small scale phenomena, such as local flows aroundstructures.
This paper has a twofold purpose. One is to clarify the suitable computational method for atarget phenomenon. The other is to develop the general BVC method based on a depthintegrated model without the shallow water assumption and to verify the applicability of themethod to flow field around a cylinder measured by Roulund et al.(2005).
2.

Previous Bottom Velocity Computational Method with Shallow Water Assumption

BVC method has been developed based on equation (1), which is derived by depth integrating
vorticity.
Wh
z
z
(1)
ubi  usi   ij 3 j h  ws s  wb b
xi
xi
xi
Here, ubi: bottom velocity, usi: water surface velocity, j: depth-averaged vorticity, h:water
depth, ws, wb: vertical velocity on water surface and bottom, respectively, zs: water surface
level, zb: bottom level. For the previous method (Uchida & Fukuoka, 2009, 2010, 2011),
bottom velocity is evaluated by water surface velocity and depth-averaged vorticity,neglecting the last three terms in equation (1) by the shallow water assumption, in which theratio of representative water depth h0 to representative horizontal scale L0 is very smalls=h0/L0<<1 (s: shallowness parameter). Water surface velocity and depth-averaged vorticity
are calculated by equation (7) and (10), respectively, together with shallow water equations ofcontinuity equation, momentum equation (2) and the turbulence energy transport equation.
And hydrostatic pressure distribution, which is a kind of the shallow water assumption, isassumed to compute depth averaged velocity and water surface velocity equations. Governing
equations of the BVC method are set by adding cubic vertical velocity equation (2).
ui  ui 12 3  12 2  1  ui  4 3  3 2   U i(2)Here, Ui: depth averaged velocity, ui: usi ubi, ui: usi Ui.
3.

Applicability Ranges and Limitations of Calculation Models by Shallowness

3.1 Depth averaged velocity field

To clarify the applicability ranges and limitations of 2D and 3D models for depth averaged
velocity fields, the magnitude of each terms in depth averaged velocity equation are discussed
by shallowness parameter in this section. Equations for horizontal depth averaged velocity arederived by depth-integrating momentum equations in horizontal directions without theassumption of hydrostatic pressure distribution.
 U h U U h  gh 2 / 2 hdp0
zz  h ij  i  i j    gh b   bi  
 dpb b  (3)
x j 
xi
xi
xi  x j
 xi
 t

2

Here, bi: bed shear stress, dp: pressure deviation from hysrostatic pressure distribution(p=g(zsz)+dp), dp0: depth averaged dp, dpb: dp on bottom, ij :horizontal shear stress due toturbulence vt Sij and vertical velocity distribution ui ' u j ' , Sij: depth averaged strain velocity,ui ' : velocity deviation from depth averaged velocity.

Expressing representative horizontal scale L0 , water depth (vertical scale) h0 and velocity U0,
physical quantities in equation (3) are non-dimensionalized as
U i  (U i )*U 0 , h  (h)* h0 ,xi  ( xi )* L0 ,  0i /   ( 0i )*U 0  * , dp  (dp)*U 0  s ,
2

2

2

2

 t Sij   01.2U 0 2 * s ( t )* ( Sij )* , ui ' u j '   0 0.5 *2U 0 2 (ui ' u j ')*(4)Here, *=u*0/U0, u*0: representative shear velocity, 0=0.1. vt is evaluated by zero-equation
model, vt=u*h/6. ui ' u j ' is evaluated by vertical velocity distribution of quadratic curve(Uchida & Fukuoka, 2009). By substituting equation (4) into equation (3), equation (5) is
derived.

 
 

1
 
 
2  dp 
Fz   0   s2  Fc  2 Fh   s 2 Fdp  Fr 2 s  b   0 s Ft  0s.5 Fc  0 (5)
Fr
*
0
 *
 h  *  h 
Here, Fz=gravity terms, Fc: convection terms, Fh, Fdp: pressure terms of hydrostatic, nonhydrostatic e*=0.1
component,
F , F : horizontal shear stress term due to turbulence, vertical
e*=0.1 2 t 2 cvelocity distribution, Fr =U0 /gh0.
*

*

3

1.2

magnitude of each term
Magnitude of each term in depth
termof eachmagnitude
averaged velocity
equation

1.0 1.02Gravityandandbedbottom
shear
Gravity
stressterms
shear
stress terms

0.8 0.8

Convection
terms
Convection
terms

0.6 0.6
Non-hydrostatic
Non-hydrostaticpressureterms
pressure
terms

0.4 0.4

Reynolds
stressterms
Reynolds
stressterms

0.2 0.2
0.0 0.0-3
1.0E-0310
1.0E-03-2
1.0E-0210
1.0E-02-1
1.0E-0110
1.0E-01

1.0E+00
1
1.0E+00

1.0E+01
1.0E+01s

Horizontal
shear
stressHorizontal
shear
stressterms
duedueto verticalterms
to verticalvelocity
distribution
velocity
distribution

3D

2D

Figure 1: Magnitudeof each term in depth averaged velocity equation
Shallowness parameter es
Shallowness parameter es

Figure 1 shows magnitude of each term in equation (5) by shallowness parameter s for *=0.1,
setting the maximum magnitude for s to 1 in the vertical axis. For very shallow water flows<10-3, depth averaged velocity field is determined only by the gravity and the bed shear
stress terms without influence of the convection terms. In this range, we can apply formulae
for uniform flows such as the Chezy formula or the Mannig formula to evaluate depth
averaged velocity. The convection term is dominant for 10-3 < s < 1, in which a depth integrated
model is useful. However, the non-hydrostatic terms become larger from s=0.1. So, for 0.1 < s
< 1, a non-hydrostatic depth integrated model (quasi-3D model) is required, while 2D model isuseful for 10-3 < s < 0.1. A 3D model is required for small scale phenomena of 1 < s, in which
the convection terms become smaller with increasing s. The horizontal shear stress terms due
3

to turbulence and vertical velocity distribution are small regardless of shallowness parameter,
because the magnitudes of these terms are evaluated for uniform flow condition in this study.
3.2 Bottom velocity field

A calculation method of bottom velocity has an important role for a bed variation analysis. A
non-dimensional equation of (1) isub   us    0 0.8 * h   s 2 Fw
(6)Here, Fw: spatial difference terms in vertical velocity of equation (1).
The equation concerning a water surface velocity is derived from momentum equation in thevery thin layer zs under the water surface (Uchida & Fukuoka, 2010, 2011).
us i
u
z
 usj s i   g ' s  Psi
(7)
t
x j
xi
2vt
12u sei  u si   3ui  6ui (8)h2Here, g '  g  (1 /  )(dp / z) , Psi :shear stress acting on undersurface of zs, usei = Ui + (uiPsi ui), ui: usi ubi, ui: usi Ui. According to equation (4), non-dimensional equation forwater surface is given byu s *  u se *   0  s Fsc  Fs  Fh2 (1  Fr 2 s 2 Fsr )
0.2

(9)
Fr


Here, Fsc: convection terms, Fs: shear stress terms due to non-equilibrium vertical velocity
distribution, Fsr: curvature terms of water surface.

*A depth averaged horizontal vorticity equation is derived from depth integrated horizontal
vorticity equation (Uchida & Fukuoka, 2009, 2010, 2011).
hDiji h ERi  P i 
(10)
t
x j
Here, ERi  usis  ubib , P i  C p vtb (bei  bi ) / h, C p   /  ,    / 6, vtb: vortex
viscosity on bottom converted to depth averaged value,v iDij  U j i  U i  j   ' j u 'i   'i u ' j  t  x j
The depth averaged vorticity equation (10) is non-dimentionalized ash*  e h*   *1 0 0.1 s F1   * 0 0.8 F 2   01.2 * s F 3 (11)Here, F1, F2 and F3 are vorticity transport terms due to depth-averaged, vertical velocity
distribution and turbulent diffusion.
Substitution of equation (9) and (11) into equation (6) leads to equation (12) for bottom
velocity pattern.
ub   U *   0 0.2  s Fsc  Fs  Fh2 1  Fr 2 s 2 Fsr 
* 
Fr

(12)
s
1.2
2
 0.9 F1   * F 2   s  * 0 F 3   s Fw



0







4

Figure 2 shows magnitude of each term in the bottom velocity equation (12) for *=0.1. Fors<10-2, because bottom velocity field is determined only by depth averaged velocity field, a
2D model is applicable to evaluate bottom velocity for bed variation analysis. On the other
hand, for 10-2 < s, the convection terms of water surface velocity and depth averaged horizontal
vorticity become larger with increasing s. So, one cannot apply a 2D model to bottom velocity
analyses for 10-2 < s, ignoring variations in vertical velocity distribution. The upper limit s=10-2
of the applicable range of a 2D model in Figure 2 is smaller than that of Figure 1 (s=10-1). The
results indicate that a quasi-3D model plays a more significant role in bed variation analysis.
For 0.1 < s, because the magnitude of spatial differenceterms in vertical velocity becomes
*=0.1
*=0.1larger with increasing s, the shallow water assumption is unsuitable. For 1 < s , the depth
averaged velocity has little influence in bottom velocity field any more.
Depth-averaged
velocity
Depth-averaged
velocity
termterm

0.8 0.8

Convection
terms
of of
Convection
terms
water
surfacevelocitywater
surfacevelocity

0.6 0.6

Vorticitytransport
terms
Vorticitytransport
terms
due toconvectiondue
to convection

Impact on bottom velocity
Impact on bottom velocity
Magnitude of each term in bottom
velocity equation

1.0 1.0

Vorticitytransport
terms
Vorticitytransport
terms
due toverticaldue
to vertical
distribution
distribution
Vorticitytransport
terms
Vorticitytransport
terms
due toturbulencedue
to turbulence

0.4 0.4
0.2 0.2

0.0 0.0-3

1.0E-03
1.0E-0310-2

1.0E-02
1.0E-0210-1

1.0E-01
1.0E-0110

1.0E+00
1.0E+00
12D

s

1.0E+01
1.0E+01

Spatialvariation
termtermin inSpatialvariation
verticalvelocity
verticalvelocity

3D

Shallowness
parameter
es term
Shallowness
parameter
es in bottom velocity equation
Figure2: Magnitudeof eachThe above results of Figure 1 and 2 indicate that a depth integrated model without assumptionof shallow water flow such as non-hydrostatic pressure distribution and spatial difference invertical velocity is required for 0.1 < s < 1.
4.

General Bottom Velocity Computation Method without Shallow Water Assumption

4.1 Governing Equation for Depth Averaged Vertical Velocity Equation

To calculate bottom velocity by equation (1) without the assumption of a shallow water flow
condition, a Poisson equation for time variation in depth integrated vertical velocity is derived
as follows. Unknown quantities in the BVC method are computed by time advance method.
Unknown quantities at n+1 step are required to satisfy

ui n1   ij 3 ( j h) n1 

n1

n1

(Wh) n1
n1 z sn1 zb
 ws
 wb
xi
xi
xi

(13)

Here, predicted velocity difference between water surface and bottom velocity ui p byprevious depth integrated vertical velocity Whn at n step is introduced:
n1

n 1

(Wh) nn1 z sn 1 zbui   ij 3 ( j h) 
 ws
 wb
xi
xi
xi
By subtracting equation (14) from equation (13), we have
p

n1

5

(14)

 (ui ) 


n 1
p
,  (ui )  ui  ui ,   Whn 1  Whn
xi

(15)

On the other hand, corrected vertical velocity (Wh)=(Wh) n+1 Wh)p is represented as follows.
(Wh)p is defined as depth integrated vertical velocity by using predicted horizontal velocity(14). Vertical velocity at arbitraly depth =(zsz)/h) is
z

z
 

(16)
w  w  u j, w  
u
dz
j

x j
x j  zbAnd the vertical velocity distribution is corrected by (ui) :(17) ui    ui  4 3  3 2 
By substituting equation (17) into equation (16) and making depth-integration, correctedvertical velocity (Wh) is obtained ash 2 u j  Wh  k1
, k1  1/ 20
(18)
x j
Finally, the Poisson equation for vertical velocity variation in time is derived fromsubstitution of equation (15) into equation (18).
  2  
(19)
k1
h
  P    0,  P  (Wh) P  (Wh) n
x j  x j 
4.2 Governing Equation for Non-Hydrostatic Pressure Distribution

Depth integrated vertical momentum equation is used to determine non-hydrostatic bottompressure distributions.
h zjzhW hWU j dpb  bj b 
(20)
t
x j

x j
x j
For simplicity, the unsteady and horizontal shear stress terms are neglected and the linear
vertical pressure distribution is assumed:
dp  dpb(21)
5.

Application to Flow Fields around a Cylinder and Discussion

The general BVC method without the shallow water assumption is applied to a flow around acylinder mounted on fixed rough bed, which was measured by Roulund et al.(2005). Table 1
shows the experimental conditions and Figure 3 shows computational gird around a cylinder
made of 1/40 of the diameter. Experimental discharge is given at upstream end and water
depth is given at downstream end.
Table 1: experimental conditions
by Roulund et al.(2005)
Diameter D (m)Depth h0 (m)Cannel width B (m)Fr number

0.536
0.54
4.0
0.14

2
1
0
-1
-2
-4

-2

0

2

Figure 3: Computational grid

6

4

(m)

Figure 4 shows computed results of (a) depth-averaged vertical velocity, (b) non-hydrostatic
bottom pressure and (c) bottom velocity distributions. Downward and reverse bottom flows
are induced with low pressure in front of a cylinder and high pressure in both sides of acylinder. Those are typical characteristics of a horse shoe vortex. Figure 5 shows comparison
between experimental and calculational results by the general BVC method for velocity
distributions on longitudinal cross-sections along the center line of a cylinder. It isdemonstrated that the method can simulate 3D flow field in front of a structure despite of thedepth integrated model.
dpb(m/ /ss))
uubibiuubi(mbi / s()m(m)
0.10.20.20.30.30.40.40.5
0.5ubiubi
0 0 0.1
0.10.20.20.30.30.40.40.50.5
0 0 0.1
g -0.008-0.008
0 0.1 0.2
0.008
0 0.1 0.2 0.3 0.4 0.5
0 0.30 0.4 0.50.008

W (m / s)-0.12 -0.12-0.04-0.040.04
0.04 0.12
0.12
0.5

0.6
0.6 0.6

0.4
0.4
0.4

0.4

0.2

0.2

0.5

0.4
0.4

0.2 0
0.2

0

Y
Y Y

0

0.2
0.2

0

0

Y
Y

0

Y

0.5

0.6
0.6
0.6
0.5

00

-0.2

-0.2

-0.5
-0.2
-0.2

-0.5
-0.5

-0.2
-0.2

-0.4

0

-0.5

00

-0.5

0

-0.4

-0.5
-0.4
-0.4
-0.6
-0.2

2

-0.5 -0.4
-0.6

-0.8

00

-0.8

X

-0.6
-0.6

-0.6

-0.4

(b) Non-hydrostatic
bottompressure-0.8 -0.6-0.6 -0.4-0.4 -0.2
-0.2
-0.8

-0.2

0

X

-0.6
-0.6

(c) Bottom velocity

00

-0.8
-0.8

-0.6
-0.6

-0.4
-0.4

-0.2
-0.2

00

1

X to experimental
X0.4
XX
U/U0:results-0.6by-0.4-0.2 BVC0 0.2
0.6 0.8 condition
1 1.2 by Roulund
Figure 4: Calculated
the general
methodetal.(2005)

1

1.2

Calculated

X/D
-1

-0.5

0

x/D
0.5

1

X/D

0.5

7

1

1.5

1.5
Z/H

0

2

2

-1

-0.2
-0.4

X/D

-0.5

1.5

0.0

Figure 5: Velocity distribution on the central vertical-section in front of a cylinder.

-1

0

1

0.2

-0.5

0.4

-0.6

-1.5

-1.5

0.5

1

0.5

1

-1.5

1

0.2 0.4 0.6 0.8

0

1.2 0.6

1 X/D
ui/U0
1.5
2

1.2

0.5

1

2
0

0

0

0.8

0.5

0.5
-0.5

-0.5

1.0

1

-1

Measured(Roulund etal.,2005)
0.2 0.4
0.6 0.8

0U/U0: -0.6 -0.4 -0.2 0
1
-1.5
-1
1

1.2

U/U0: -0.6 -0.4 -0.2 0

Z/H

-1.5

1
ui/U0

1.2

0.5

0.2 0.4 0.6 0.8

Z/H: -0.6 -0.4 -0.2 0

11

ux/U0：

0: -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8



1.5

0

1

-0.5
(a) Depth-averaged
vertical velocity-0.4-0.6 -0.4

6.

Conclusions

Applicability of various computational methods for a target phenomenon is discussed basedon momentum equations. It is clarified that a depth integrated model without the assumptionof shallow water flow is required to calculate both of depth averaged and bottom velocity
flow fields for 0.1 < s < 1.
To develop the general bottom velocity computation (BVC) method without shallow water
assumptions, the Poisson equation for vertical velocity variation in time is derived. And
computation method for non-hydrostatic pressure distribution by using the depth-integrating
vertical momentum equation is presented.
The general BVC method is applied to flow field around a cylinder mounted on the fixed andrough bed. It is demonstrated that the general BVC method can calculate 3D flow structures
around a cylinder through the comparison with the experimental results of Roulund et al.
(2005).
References
Fukuoka, S., Watanabe, A. and Nishimura, T.(1992) On the groin arrangement in meanderingrivers, Proceedings of the Japan Society of Civil Engineers, No.443/II-18，pp.27-36,
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Nishimoto, N., Shimizu, Y. and Aoki, K.(1992) Numerical simulation of bed variationconsidering the curvature of stream line in a meandering channel, Proceedings of theJapan Society of Civil Engineers, Vol.456/II-21，pp.11-20, in Japanese.
Roulund, A., Sumer, B. M., Fredsøe, J. and Michelsen, J.(2005) Numerical and experimental
investigation of flow and scour around a circular pile, Journal of Fluid Mechanics,
Vol.534, pp.351-401.
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shallow water equations and horizontal vorticity equations, 33rd IAHR Congress: Water
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