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4 Numerical Simulation of the Viscous Flow Past a Square Prism with Attached Frontal Plates

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





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)



338



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.



340



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



342



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



344



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.



346



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.



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