4 Numerical Simulation of the Viscous Flow Past a Square Prism with Attached Frontal Plates
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17 Flow Control Near a Square Prism with the Help of Frontal Flat Plates
337
Fig. 17.6 The scheme of
discretization of the vorticity
field
In the present numerical realization of the vortex method, the finite volume discretization of the vorticity field is performed. The volumes are connected with node
points of the orthogonal grid put on the calculation domain (Fig. 17.6). The point
vortices located in the middle of each volume are characterized by the vorticity ωi j ,
where i = 1, 2, . . . , N x , j = 1, 2, . . . , N y , N x , N y are the numbers of grid cells in x
and y directions, respectively. From the divergence theorem, the law of conservation
of vorticity in the elementary volume can be described in the form:
∂
∂t
Ω
ωi j dq = −
∂Ω
ωi j (V · n)dl,
(17.19)
where Ω, ∂Ω is the discrete volume and its boundary, respectively, n is the normal
to ∂Ω and V is the flow velocity on ∂Ω. As Eq. (17.19) defines the vorticity convection across the elementary volume, we obtain the following numerical scheme for
Eq. (17.17):
ω it+Δt
− ω it j
j
t
t
t
t
ΔxΔy ≈ (ω i−1
j u i−1 j − ω i+1 j u i+1 j )Δy+
Δt
(ω
t
t
i j−1 vi j−1
−ω
t
t
i j+1 v i j+1 )Δx
−ω
t
t
i j (|u i j |Δy
+ |v
(17.20)
t
i j |Δx),
where Δx and Δy are the steps of space discretization in x and y directions, and Δt
is the time step.
It is obvious that scheme (17.20) has the first order in time and the second order
in space. Development of this approach on multilayer templates is presented in [27].
Note the scheme is dissipation-free and has improved dispersion properties compared
with classical linear schemes.
To simulate the viscous diffusion process, we integrate Eq. (17.18) by the finitedifference method. The scheme of the second order in space written in the nodes of
orthogonal grid takes the form:
− ω it j
ω it+Δt
j
Δt
=
1
Re
t
t
t
ω i+1
j − 2ω i j + ω i−1 j
(Δx)2
+
ω it j+1 − 2ω it j + ω it j−1
(Δy)2
.
(17.21)
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I.M. Gorban and O.V. Khomenko
Discrete Eqs. (17.20) and (17.21) are integrated in time with applying the explicit
scheme of the first order. Notice it is stable at the Courant numbers that do not
exceed 1.
So, that way looks to changing in time the circulation Γ i j (t) = ω i j (t)ΔxΔy of
the vortex particle fixed in the grid node unlike the classical vortex method [21, 22]
that deals with translation of free discrete vortices in the flow field. Adaptability
of the scheme is reached because of the grid points whose circulation satisfies the
condition |Γ i j | < ε, where ε is the small value, are only considered.
The Lighthill’s mechanism of vorticity creation at a solid wall and linking it to
vortex methods are described in detail in [21]. It explains the vorticity generation by
changing the circulation γ of the vortex sheet simulating the body surface because
of vorticity field modifications. In the numerical schemes of a vortex type, there are
different approaches to calculation γ and its incorporation in a boundary condition for
vorticity. We determinate the intensity of body sheet from no-through flow boundary
condition (17.3), which leads to the following integral equations with respect to γ :
Σ
γ (r , t)
∂G(r, r )
dl(r ) +
∂n
ω(r , t)
S
∂G(r, r )
ds(r ) = 0,
∂n
(17.22)
where r ∈ Σ.
The Kelvin’s theorem of circulation conservation in the computational domain
must be also satisfied:
Σ
γ (r , t)dl(r ) +
ω(r , t)ds(r ) = 0,
(17.23)
S
No-slip condition (17.4) is used to derive a boundary condition for vorticity. Taking
into account the velocity jump across the vortex sheet, one has the following relation:
(Vτ )− = Vτ0 +
γ
,
2
(17.24)
where Vτ0 is the tangential velocity of body-surface points calculated from (17.16) and
(Vτ )− is the limiting value of tangential velocity at the body, which condition (17.4)
has to be satisfied for. Following Wu [28] who divided the strength of the vortex sheet
by the distance from the wall to the first mesh point in the computational domain
to obtain the vorticity on the body, we get the Dirichlet-type boundary condition for
vorticity in the following form:
2Vτ0
,
(17.25)
ω0 =
Δs
where Δs is the grid spacing perpendicularly to the wall.
The vorticity created on smooth walls enters the fluid through a mechanism of
viscous diffusion described by formula (17.21). And the sharp edge vorticity is
transferred to the flow with applying convection formula (17.20) that is equivalent
to implementation the Kutta–Joukowski condition in this point.
17 Flow Control Near a Square Prism with the Help of Frontal Flat Plates
339
17.4.2 Calculation of the Pressure Field and Forces
on the Body
The introduction of vorticity and velocity–vorticity formulation of the Navier–Stokes
equations allow to decouple purely kinematical problem from the pressure problem.
It simplifies significantly numerical modeling of the hydrodynamic fields. But to
estimate either the fluid forces acting to a body or sound level in the flow, one is need
of calculating the pressure at least on the body. It has to be noted that recovery of the
pressure from vorticity and velocity fields is a daunting challenge, which has invited
attention of many researchers [29–31]. When direct solving the Poisson equation
for the pressure, the problem of the correct choice of boundary condition arises. On
the other hand, use of alternative approaches such as variational formulation [29]
or Uhlman’s integral [30] is difficult due to having sharp edges in the considered
geometrical configuration.
We derive the pressure field by direct integrating the Navier–Stokes equations in
the Lamb representation [32]:
1 ∂p
1 ∂ 2
1 ∂ω
∂u
+
(u + v2 ) − νω = −
−
,
∂t
2 ∂x
ρ ∂x
Re ∂ y
(17.26)
1 ∂ 2
1 ∂ω
1 ∂p
∂ν
+
(u + v2 ) + uω = −
+
.
∂t
2 ∂y
ρ ∂y
Re ∂ x
(17.27)
It is obvious Eqs. (17.26) and (17.27) connect the pressure field with velocity and
vorticity fields. Integrating Eq. (17.26) of the variable x and Eq. (17.27) of the variable
y, one obtains the following formulae for calculation the dimensionless pressure:
p = 1 − u 2 − v2 + 2
p = 1 − u 2 − v2 + 2
y
−∞
x
−∞
vω −
−uω −
∂u
1 ∂ω
−
d x,
∂t
Re ∂ y
∂v
1 ∂ω
+
dy,
∂t
Re ∂ x
(17.28)
(17.29)
2
where p = 2( p − p∞ )/ρU∞
.
It depends on the flow field configuration, what equation from (17.28) and (17.29)
will be chosen for calculating the pressure. Note that this way allows deriving the
total drag including its form and viscous components.
The coefficients of fluid forces on the body are calculated using the pressure
distribution:
pn x d x, C y =
pn y dy,
(17.30)
Cx =
L
L
where C x , C y are the coefficients of drag and lift, respectively, and n = (n x , n y ) is
the internal normal to the body.
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I.M. Gorban and O.V. Khomenko
Fig. 17.7 Sketch of the
computational grid
In the present numerical scheme, Eqs. (17.28) and (17.29) are integrated with the
trapezium method on the base orthogonal grid.
17.4.3 Validation of the Algorithm
With the vortex method described above, the present simulation results for an impulsively started square prism at moderate Reynolds numbers (Re = 100 ÷ 600) are
validated against theoretical, experimental, and numerical data available in the literature. In this study, we adopt the three-level rectangular grid with a constant cell size
at each level as presented in Fig. 17.7. The grid spacing Δ1 in the domain adjoining
the body coincides with the length of the panels that simulate the bound vortex sheet.
And the cell size of each next grid is doubled compared with the previous. The number of the nodes throughout the square side is determined after preliminary tests as
Ns = 50 that leads to Δ1 = 0.02. The dimensionless width of the calculation region
is 20 and the lengths of upstream and wake regions are 10 and 90, respectively. For
all the cases investigated in this paper, the normalized computational time step is
equal to Δt = 0.01.
Figure 17.8 presents the variation of Strouhal number St and mean drag coefficient C D with Reynolds number for square prism from the present simulations. The
shedding frequencies were determined from the power spectra of the nonstationary
lift signals, as well as velocity fluctuations in the wake.
Included for comparisons are the known experimental data of Okajima [33, 34]
together with the results of 2D and 3D DNS simulations of Norberg et al. [23]. In spite
of the fact that 3D effects develop in the square cylinder flow starting from Re ≈ 170
[23], the present results are seen to be in close agreement with the experimental data
and in reasonable agreement with the numerical results. Note, when Re ≥ 150, the
mean drag coefficient obtained matches as experimental as numerical results very
good. At the same time, the Strouhal numbers predicted by 3D simulations are
17 Flow Control Near a Square Prism with the Help of Frontal Flat Plates
341
Fig. 17.8 a Strouhal number St, b mean drag coefficient C D of a square prism against Re
not necessarily more “accurate” than the present results. Generally, the performed
comparisons indicate the good correlation of both the time-mean drag and shedding
frequency calculated with known experimental and numerical data.
As for quantitative characteristics of the lift force acting to a square cylinder, those
are scarce in the literature. Table 17.1 contains data for the root-mean-square value
(i. e., standard deviation) of the lift coefficient C L r ms obtained in the present calculations and known from previous researches at Re = 150 and Re = 500. Among
those, data from [27] are only experimental and all other are acquired in numerical
simulations. The coefficient C L r ms has been shown to be extremely sensitive as to Re
variations as to aspect ratio of the computation domain [23, 27] that explains significant discrepancies in the results. Nonetheless, the present C L r ms values compare
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I.M. Gorban and O.V. Khomenko
Table 17.1 Comparison of lift force standard deviation C L r ms for a square cylinder at Re = 150
and Re = 500
Source
C L r ms
Re = 150
Re = 500
Ali et al. [4]
Sohankar et al. [23]
Doolan [35]
Shimizu et al. [36]
Hwang and Sue [37]
Present simulation
0.28
0.23
0.296
–
–
0.23
–
1.13–1.22
–
0.56–0.72
0.9–1.01
0.9
reasonably well with other numerical results at Re = 150 and are in good agreement
with experimental data at Re = 500. The performed comparisons indicate that the
present version of the vortex method is able to predict correctly the flow past a square
prism at moderate Reynolds numbers.
17.4.4 Square Prism with Attached Frontal Plates. Results
of Simulation
In this section, an effect of two symmetrical plates attached to the prism frontal side
on flow structure and prism loads is studied. As we consider the possibility of small
control impact upon the flow, the plates are quite short and thin. The normalized plate
length and width are l = 0.2 and w = 0.02, respectively. A plate position toward the
adjacent prism edge r is chosen from the dependency presented in Fig. 17.5b, which
has been obtained in the previous section by the reduced order model. That has to
guarantee a stable recirculation zone between the plate and the prism frontal side.
Here, the value of ropt corresponding to the chosen plate length is 0.16.
An effect of the plates is as early as obvious if one compares the flow patterns
developed beyond a square prism without control and under the optimal control. In
Fig. 17.9, we present the vorticity fields obtained in the uncontrolled flow at Re = 150
(Fig. 17.9a) and Re = 500 (Fig. 17.9b). In all the figures, solid and dashed lines represent positive and negative vorticity values, respectively. At Re = 150, which is still
before the onset of 3D effects, the wake is seen to be laminar, regular and characterized by the primary instability, the von Karman vortices. The estimated Strouhal
number characterizing the vortex shedding frequency is 0.145, which is close to
the experimental data of Okajima [33] (St = 0.148) and slightly smaller than the
computational value of Inoue et al. [38] obtained by high-order direct numerical
simulations (St = 0.151). As regards calculations at Re = 500, those have approximation character because the transition to 3D flow behind the prism occurs well
before, at Re ≈ 190 [23]. In particular, significant levels in components of nonspanwise vorticity can be presented near the body at Re = 500. However, the flow patters
17 Flow Control Near a Square Prism with the Help of Frontal Flat Plates
343
Fig. 17.9 Vorticity contours past a square prism without control at a Re = 150 and b Re = 500:
solid line—positive circulation, dashed line—negative circulation
Fig. 17.10 Vorticity contours past a square prism with optimal control at l = 0.2, r = 0.16, a
Re = 150 and b Re = 500
and loads obtained in our calculations are close to those observed in nature. So, the
Strouhal number is 0.135 that coincides with experimental data of Norberg [39].
Other characteristics are also in good agreement with experimental and numerical
data available in the literature that is shown in Fig. 17.8b and Table 17.1.
Figure 17.10 illustrates the wake patterns generated beyond the prism with the
attached frontal plates (l = 0.2, r = ropt = 0.16). Note the vorticity contours in
Figs. 17.9 and 17.10 correspond to not only identical Reynolds numbers but also
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I.M. Gorban and O.V. Khomenko
an identical instant when the flow with or without the control plates is well established. The structure of separated flow in Fig. 17.10a, b is seen to be different from
those observed without the control. The wake width becomes narrower and much
more regular, especially at Re = 500. The vortex shedding period in the controlled
flow decreases as compared to the natural prism flow that results in the reduction
of both the intensity and the scale of the wake vortices. An influence of the control
plates on the flow grows significantly when increasing the Reynolds number. The
obtained nondimensional frequency of vortex shedding (Strouhal number) is 0.157
at Re = 150 against St = 0.195 at Re = 500. It means the increase of St in comparison with the natural frequency is 8 % in the first case and more than 40 % in the
last case.
It is shown in Fig. 17.10a, b the prism front lies inside the recirculation zones
generated by plate ends. The phenomenon as well as lowering the recirculation
bubble length and wake realignment causes drastic redistribution of pressure over
the body. Figure 17.11a, b compares the time-averaged pressure coefficient C p =
2
over the prism calculated without control plates and with the plates
2( p − p∞ )/ρU∞
at Re = 150 and Re = 500, respectively. The pictures demonstrate equalizing the
pressure at the frontal side and increase of the base pressure coefficient C pb in the
Fig. 17.11 The pressure
coefficient over the prism
surface without control
(dashed line) and with
optimal control (solid line) at
a Re = 150 and b Re = 500
17 Flow Control Near a Square Prism with the Help of Frontal Flat Plates
345
Fig. 17.12 Instantaneous drag coefficient C D and lift coefficient C L without control (curves 1),
with optimal control (curves 2), with nonoptimal control (curves 3) at a Re = 150 and b Re = 500
controlled flow. At Re = 150, the base pressure coefficient rises from C pb = −0.73
in the natural flow to C pb = −0.6 in the controlled flow and at Re = 500, the increase
is from C pb = −1.2 to C pb = −0.8. It is obvious the tendency leads to decreasing the
prism drag, which is expected to be more significant at Re = 500. Note the obtained
values of C pb in the natural flow are close to DNS data of Sohankar et al. [23] that
is important for the verification of our numerical scheme.
So, the attached frontal plates significantly affect both flow pattern and pressure
distribution about the prism and one can thus expect change of the fluid forces as
compared to the uncontrolled flow. Figure 17.12a, b shows the temporal traces of the
drag (C D ) and lift (C L ) coefficients of square prism for uncontrolled and controlled
flows at Re = 150 and Re = 500, respectively. Here curves labeled 1 correspond to
the natural prism flow, curves labeled 2 describe the prism characteristics at optimal
control (l = 0.2, r = 0.16), and curves as 3 deal with nonoptimal control when plate
position r is chosen independently of the results obtained with applying the standing
vortex model.
The presented data demonstrate substantial reduction of the hydrodynamic loads
of the prism with attached frontal plates. The mean value of the prism drag coefficient
at Re = 150 derived from its time history in our simulation (curve 1 in Fig. 17.11a) is
C D = 1.4 that coincides exactly with both experimental [34] and DNS [23, 38] data.
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I.M. Gorban and O.V. Khomenko
The prism modification with optimal parameters of the control plates is obtained to
reduce the coefficient up to C D = 1.08, curve 2 in Fig. 17.11a. So, drop in drag force
is about 22 % in comparison with the uncontrolled flow. At Re = 500, lowering the
prism mean drag is more essential; here, the coefficient C D decreases from value 1.76
obtained without control up to 1.12 that corresponds to optimal control. The control
effect is about 35 % in this case. Analogous results are achieved for fluctuating forces
acting on a square prism. The amplitudes of both the drag and the lift coefficients
decrease significantly in the controlled flow. For example, at Re = 500, the amplitude
of C L in the controlled flow is a third of that in the uncontrolled flow.
To emphasize an importance of the results derived by the simplified model of
a standing vortex, we carried out the simulation with nonoptimal parameters of the
control device. Curves 2 in Fig. 17.11a correspond to the case when the plate position
r = 0.22 that exceeds the optimal value ropt , so the plates are located too far from
the prism edges. On the contrary, the case presented in Fig. 17.11b is characterized
by too close displacement of the plates in respect to the prism edges, here r = 0.08.
In both configurations, reducing mean drag and fluctuating forces are seen to be less
than at the optimal ratio of plate length to its position.
The plots of instantaneous streamlines presented in Fig. 17.13 interpret the possible flow topology around the prism with the control plates. The results were obtained
in the numerical simulations at Re = 500. Three snapshots of these streamline plots
Fig. 17.13 Streamlines around the square prism with control plates at plate length l = 0.2, Re =
500 and different plate position: a r = 0.22, b r = 0.08, c r = 0.16
17 Flow Control Near a Square Prism with the Help of Frontal Flat Plates
347
Fig. 17.14 Strouhal number
of the square prism without
control (fill squares) and with
control (fill circles) against
the Reynolds number
correspond to an identical instant inside the period of vortex shedding and differ by
the plate location in respect to the prism edge. In the first mode, when r > ropt , the
streamline get off the plate end meets the prism at the frontal side (Fig. 17.13a). The
small vortex restricted by zero streamline weakens but cannot prevent in full the flow
separation in the prism leading edge. In the second case, no streamline reattachment
to the prism is observed (Fig. 17.13b). Only the last picture derived at r = ropt demonstrates reattachment of zero streamline close to the prism leading edge (Fig. 17.13c).
The recirculation zone restricted by the streamline is stable enough and able to suppress the flow separation at the prism edge. As a result, the recirculation bubble in
the rear of prism is smaller than in previous modes. The conclusions are identical to
those that have been made with applying the standing vortex theory (Fig. 17.4). This
fact stresses an importance of development of simple topological models in order to
forecast optimal properties of the devices used for a flow control.
Figure 17.14 shows the modification of the Strouhal number St by the frontal
plates at its optimal configuration (l = 0.2, r = 0.16) for different Reynolds numbers. For comparison, values of St in the uncontrolled flow are also represented.
Monitoring the vortex shedding frequency is known to be one way to quantify the
processes occurring in the body wake. So, an increasing of the prism Strouhal number in the controlled flow indicates the substantial change of the wake pattern as it
has been shown in Fig. 17.10. It follows from Fig. 17.14 the changes are typical for
all Reynolds numbers from the considered range. The plate effect on the shedding
frequency is seen to grow when increasing the Reynolds number.
Figure 17.15 depicts the mean drag coefficient C D against the Reynolds number
obtained at the optimal plate control. For the range of Reynolds number considered
in the present study, the coefficient C D is seen to reduce greatly in the controlled
flow. At the same time, its change with the Reynolds number is rather weak. It points
out the fact the relative decrease of the drag force under the control is higher at large
Reynolds numbers.