Study of Forced Boundary Layer by Means of DNS

International Journal of Mathematical Modeling, Simulation and Applications ISSN: 0973-8355 Vol. 3. No. 4, 2010 pp. 347-365

Study of Forced Boundary Layer by Means of DNS A. Zarghami, M. Feizabadi Farahani and M.J. Maghrebi Department of Mechanical Engineering Shahrood University of Technology 7th Tir Sq. – Shahrood – Zip Code: 36199-95161- P.O. Box: 316, Iran [email protected], [email protected], [email protected]

ABSTRACT

The non-dimensional form of Navier-Stokes equations for two-dimensional incompressible boundary layer flow are solved using direct numerical simulation. The governing equations are discretized in streamwise and cross-stream direction using a sixth order compact finite difference scheme and a mapped compact finite difference method, respectively. An algebraic mapping of y = (Ly y0h)/(y0 + Ly (1 – h)) is used to relate the physical domain of y to the computational domain of h. The third order Runge-Kutta method is used for the time-advancement purpose. The convective outflow boundary condition is employed to create a non-reflective type boundary condition at the outlet. The numerical results show a very good accuracy and agreement with exact solution of the Navier-Stokes equation. The results of boundary layer simulation also indicate that the time-trace of the velocity components are periodic. Results in self-similar coordinate were also investigated which indicate that the time-averaged statistics for velocity, vorticity, turbulence intensities and Reynolds stress distribution tend to collapse on top of each other at the flow downstream locations. Keywords: Boundary layer, compact finite difference, DNS, self-similarity.

1. INTRODUCTION When a fluid is flowing in the vicinity of the solid boundary, the effect of no slip condition appears in the form of shear stress. By influencing of this effect, a thin layer is created in the flow called boundary layer. Within this layer the velocity is changing from zero to the velocity of potential flow out of the boundary layer [1]. In recent years the techniques of computational fluid dynamics have been used to compute flows associated with geometrically complex configurations. However, success in terms of accuracy and reliability has been limited to cases in which the effects of turbulence and transition could be modeled in a straightforward manner. Even in simple flows, the accurate computation of skin friction and heat transfer using existing turbulence models has proved to be difficult task, one that requires extensive fine-tuning constants in the turbulence models used. In complex flows such as turbomachinery, the development of a turbulence model that accounts for all scales of turbulence and predicts the onset of transition may prove to be impractical. Fortunately, current trends in computing suggested that may be possible to perform direct simulation of turbulence and transition at moderate Reynolds number in some complex cases in the near future [2]. With advent of large-

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scale computers, there has been a veritable explosion of numerical work done in concert with the experiments in an effort to understand the physics of shear layers. Generally, there are two advanced methods of computing turbulence flow: large eddy simulation (LES) and direct numerical simulation (DNS). In LES, a low-pass spatial filtering is applied to the Navier-Stokes equation and the filtered equations are solved directly. It is most promising for engineering flow of low-to-moderate Reynolds numbers [3, 4]. A review of LES for incompressible flows can be found in [5]. Since the governing equations are directly solved without the use of any modeling, the solution method is called direct numerical simulation (DNS). The main purpose of DNS is to solve (to best of our ability) for the turbulent velocity field without the use of turbulent modeling. This condition means that the Navier-Stokes momentum equation for fluid must be solved exactly, which is not a simple task [2, 6]. Thus, any DNS code is very time-consuming and the extensive storage requirements. The DNS requires a large number of grid points and time steps to reach a statistically steady state and are usually limited to relatively low Reynolds numbers. With the advantage of powerful supercomputers, numerical simulation have become a viable tool for investigating boundary layer flows such as thermal boundary layer [7] and turbulent boundary layer with separation [8]. Recently, DNS by the accurate finite-difference method are available for the simulation of turbulent flow on a complex configuration [9, 10], and DNS of boundary layer flows are used for elucidating the detail structures of near-wall turbulence [11, 12, 13]. In contrast to its incompressible counterpart, DNS of compressible turbulent flow has been fairly recent. The DNS of homogeneous compressible turbulence was initiated in 1981 by Feiereisen et al. [14], but a serious study of compressible homogeneous turbulence (isotropic and sheared) was undertaken only a decade later [15]. In this paper, we report on a direct simulation of turbulence in a spatially evolving boundary layer on a flat plate. The governing equations are derived from the full incompressible NavierStokes equations. These are solved in a domain which is finite in the streamwise direction, x and semi-infinite in the cross stream direction of y. In the x direction, a high order compact finite difference scheme is used. In the y direction, a mapped compact finite difference method is employed. All quantities are non-dimensionalized by the appropriate characteristic scales of the flow. Specially, all lengths are normalized by the boundary layer thickness, d and velocities are normalized by U¥, where U¥ is the free stream streamwise velocity. Those parameters pertaining to time are normalized by U¥.

2. THE GOVERNING EQUATION Figure 1 shows the coordinate system and the computational domain in which the governing equations for the incompressible boundary layer flow are solved. The inlet velocity profile is specified by U0 (y) that has a superimposed computational velocity. The boundary layer flow is allowed to develop in the spatial direction x. Applying Newton’s second law of motion for a Newtonian fluid particle gives the equation of motion, known as the Navier-Stokes equations. These equations together with an equation representing mass conservation are the governing equation for an incompressible boundary layer flow.

A. Zarghami, M. Feizabadi Farahani et. al.

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U¥

U¥ y

d(x)

U(x, y)

x

Fig. 1

Spatially developing boundary layer geometry.

The governing equations (Eq. (1) and Eq. (2)) have been non-dimensionalized by the characteristic length (d) and velocity scales (U¥). H H H H ∂U 1 (1) + (U. ∇)U = – Ñp + (∇ 2U ), ∂t Re Ñ.U = 0. (2) One of the main difficulties in the solving the Navier-Stokes equations is the lack of information about the pressure at the boundaries. This is overcome by either using the staggered grid for the discretization or by eliminating the pressure term from the Navier-Stokes equations. Application of the following vector identity: Ñ (A.B) = (B.Ñ)A – (A.Ñ) B – B ´ (Ñ ´ A) – A ´ (Ñ ´ B) (3) H for the case of A = B = U = (U, V, W) results in the formation of Eq. (4) H H H H 1 H H (7 . ∇ )7 = ω × U + ∇(U. U ) (4) 2 H where ω = (w1, w2, w3) = Ñ ´ U. If Eq. (4) is substituted for the convective term in the nondimensionalized Navier-Stokes equation (Eq. (1)), it gives: H H H H H ∂U 1 U .U = H − ∇( p + (5) )+ ( ∇ 2U ) ∂t 2 Re H H H where H = (H1, H2, H3) = 7 × ω . Taking Ñ ´ (5) results in H H H H H 7 .7 ∂( ∇ × 7 ) 1 2 = ∇ × 0 − ∇ × ∇( F + ∇ (∇ × 7 ) (6) )+ ∂J 2 Re which will further simplify to Eq. (7) on the usage of Ñ ´ Ñ (scalar) = 0. H H ∂ω 1 2H ∇ ω = ∇×H + (7) ∂J Re by taking Ñ ´ (7), the following equation will be obtained. H H H ∂∇ × (∇ × U ) 1 2 = ∇ × (∇ × H ) + ∇ (∇ × (∇ × U )) (8) ∂t Re H using the continuity equation (Ñ.U = 0) and applying the next vector identity H H H Ñ ´ (Ñ ´ U ) = Ñ (Ñ.U ) – Ñ2U (9)

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converts Eq. (8) into the following equation H H ∂∇ 27 1 4 H ∇ U = ∇ × (∇ × H ) + (10) ∂J Re H H H where the vector H = U × ω contains the non-linear terms and Re = DU dw /v. Equations (7) and (10) are the evolution equations responsible for the time-advancement in the simulation. The instantaneous velocity U = (U, V ) is decomposed into a base flow (U0 (y), 0) and the computational flow velocity components (u (x, y, t), v(x, y, t)) as follow. U (x, y, t) = u (x, y, t) – U0 (y)

(11)

V (x, y, t) = v (x, y, t))

(12)

Using the streamwise components of Eq. (7) and Eq. (10) and the decomposition shown by Eq. (11) yields:

∂ 2 ∂2 ∂2 1 4 ∇ u= ∇ U H1 − H2 + 2 ∂t ∂ ∂ x y Re ∂y

(13)

where w1 = w2 = H3 = 0 for the case of two-dimensional flow. Equation (13) and the convective outflow boundary condition are responsible for the time-advancement of the simulation. The crossstream velocity component n is recovered directly from the continuity equation.

∂u ∂v . =− ∂x ∂y

(14)

The vorticity component w3 is calculated following its definition. w3 =

∂V ∂U + . ∂y ∂x

(15)

3. BOUNDARY AND INITIAL CONDITIONS Equation (13) is a fourth-order, partial differential equation, so it requires four boundary conditions. The u velocity is specified at the inlet (x = 0) and the outlet boundaries (x = Lx ). With the help of continuity equation, ¶u/¶x is also specified at the inflow and outflow boundaries:

∂u ∂L =− . ∂O ∂x

(16)

The former and the latter are known as Dirichlet and Neumann type boundary conditions, respectively. The boundary conditions are set to zero in the transverse direction. In the numerical simulations, the instantaneous velocity component at the inlet plane of domain, termed “reference” velocity also, is represented by: U(y) = 0.322 y – 0.00023 y4 + 1.998 ´ 10–3 y7 – 1.571 ´ 10–7 y10 + 1.13 ´ 10–9 y13 (17) or

A. Zarghami, M. Feizabadi Farahani et. al.

R|1.8y − 1.9683y || U (y) = S1 − 0.81 (1 + y ) ||1 |T

4 2

FG 0 ≤ y ≤ 4 IJ H 9K FG 4 ≤ y ≤ 1IJ K H9

!#

(18)

( y ≥ 1)

which is superimposed by some perturbations. The perturbations are introduced in the form of a traveling wave. The perturbation part, which is a combination of linear eigenfunctions obtained from the linear stability calculations, is specified for the inflow boundary condition.In other words: v(x, y, t) = A × Real [V(y) ei (– wt)]

(19)

where V (y) is the velocity eigenfunction corresponding to the most amplified mode of the twodimensional Orr-Sommerfeld equation and A is the amplitude of the two-dimensional forcing which corresponds to the fundamental frequency. Convective outflow boundary conditions are specified at the outflow. The boundary conditions must be non-reflective to avoid feedback problem. The convective boundary conditions are used to generate the Dirichlet boundary conditions for both velocity components. ∂ψ ∂ψ = −c (20) ∂J ∂x where Y is replaced by each of the velocity components. In (20), c represent the advection speed of the large-scale structures in the layer. The purpose of this condition is to allow the fluid structures to flow out of the domain in a natural manner. Therefore, the advection speed is chosen between zero and one. For the low Reynolds number flows simulated in this work however, the small-scale structures are, in fact, relatively large. Moreover, the results indicate that the region of the influence of the outflow boundary condition is restricted to a fairly short distance upstream of the exit plane-

roughly one layer thickness. Therefore, the choice of c = U is appropriate for these simulations as will be evident in the result to outflow. An unforced, two-dimensional boundary layer simulation whose inlet boundary contains a base profile (Eq. (17) or Eq. (18)), provided the initial conditions for the forced boundary layer simulations. A uniformly distributed mean velocity at all x stations is the initial condition for the unforced two-dimensional boundary layer simulation. These initial conditions must then be allowed to wash out before performing any statistical analysis on the layer. In other words, any particle at the inlet (x = 0) must be allowed to leave the outlet boundaries (x = Lx). The boundary layer flow must also reach the statistically stationary state in which the mean velocity component is time independent.

4. NUMERICAL FORMULATION The details of the numerical methods employed in the current simulations are provided here. As discussed, the spatially developing boundary layer is solved in a domain with a finite extend in the streamwise direction and semi-infinite (0 £ y £ µ) in the major-gradient (MG) direction. A mapping is employed to convert the semi-infinite y extend of the original domain into a computational

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domain of h with interval 0 £ h £ 1. Equations (13), (14) and (15) indicate that for each time step (sub-time-step) a method is required that can: • evaluate the spatial derivatives, • integrate the continuity equation to recover v from Eq. (14), • compute the non-linear terms in Eq. (13), • solve the two-dimensional Poisson equation for u. At the end of each time step the solution can be regarded as a new initial condition for u. This is required for time-advancement of the computations. The derivatives in the streamwise direction are computed using the Pade’ finite difference scheme developed by Lele [16]. He introduced the first derivative of f (x) implicitly according to: 4α − 1 α+2 (21) ( f j+1 − f j−1 ) + ( f j+ 2 − f j−2 ) 3h 12 h where a prime denotes the first derivative, j represents the grid number (0 £ j £ J ) and h = Dx = Lx/J. By setting a = 1/4 or a = 1/3 a fourth-order or sixth-order accurate scheme is obtained. At the streamwise boundaries (e.g. at j = 0 and j = J) an implicit one-sided, third-order derivative approximation is used:

a f ¢j

–1

+ f ¢j + a f ¢j + 1 =

f ¢0 + 2 f ¢1 =

1 (− 5 f 0 + 4 f 1 + f2 ) 2h

(22)

1 (23) (5 f J − 4 f J − 1 − f J − 2 ) 2h At the vicinity of the boundaries (e.g. at j = 1 and j = J – 1) the general form of the first derivatives (Eq. (21)) is used with a = 1/4. Lele [16] discussed that replacing of a by a¢ = (16a – 32)/(40a – 1) at j = 2 and j = J – 2 guarantees the stability and numerical conservation of (¶/ ¶t) u = (¶/¶N) f (u). The Pade’ finite difference scheme, as introduced above, is an implicit scheme. It is expected to realize a third order accurate scheme at the boundaries and a sixth order accurate scheme far from both boundaries (e.g. at x = Lx/2). Figure 2 shows the order of accuracy [17]. Equation (24) represents the second derivative of f (x), which is the family of fourth order accurate Pade’ finite difference schemes. f ¢J + 2f ¢J–1 =

where

4(1 − α )

f ¢¢0 + 11f 1¢¢ =

2

1

h2

( f j+1 − 2 f j + f j−1 ) +

10α − 1

( f j+ 2 − 2 f j + f j−2 ) 3h 12 h 2 a = 1/4. At the boundaries, one-sided, third order scheme are used. They are: a¢¢j – 1 + f ¢¢j + a f ¢¢j +1 =

( 13 f 0 − 27 f 1 + 15 f 2 − f 3 )

(24)

(25)

and

1 (13 f J − 27 f J − 1 + 15 f J − 2 − f J − 3 ) h2 Taking the first-order derivative from both sides of Eq. (22) gives f ¢¢J + 2f j¢¢– 1 =

(26)

−3 1 1 (27) ( − 5 f ′ 0 + 4 f 1′ + f 2′ ) = ( f 0′ + 4 f 1′ + f 2′ ) f 0′ + h 2h 2h Substituting the left hand-side of Eq. (22) (using a = 1/4) for the terms in the parenthesis of Eq. f ¢¢0 + 2f ¢¢1 =

A. Zarghami, M. Feizabadi Farahani et. al.

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(27) forms the following equation. − 3 df 3 (28) ( )x =0 − 2 ( f0 − f2 ) h dx 2h Equation (28) is used at the inlet boundary when both the function value and its derivative are known. An analogous approach is performed to specify the second derivative at the outflow boundary when both function value and first derivatives are available.

f 0¢¢ + 2 f ¢¢1 =

f ¢¢J + 2f ¢¢J– 1 =

3 df 3 ( ) = − ( f − fJ −2 ) h dx x Lx 2 h 2 J

(29)

n = – 3.052672

–3

10

n = – 4.013592

–5

|Error|

10

at x = Lx/2 at boundaries Next to boundaries

–7

10

n = – 6.002145 –9

10

–11

10

N

30

Fig. 2

100

Order of accuracy for first derivative approximation using Pade’ finite difference scheme. 10

–2

10

n = –3.000001 –3

–4

|Error|

10

10

at x = Lx/2 at boundaries

–5

–6

10

10

n = –3.999999

–7

10

–8

10

100 N

1000

Fig. 3 Order of accuracy for second derivative approximation using Pade’ finite difference scheme.

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International Journal of Mathematical Modeling, Simulation and Applications

In the immediate vicinity of the boundaries (at j = 1 and j = J – 1), the second-order compact finite difference scheme Eq. (24) is used with a = 1/10. An evaluation of the fourth derivative, which appears in the biharmonic term of Eq. (13), is performed by imposing the second-order derivative operator twice in succession. Figure 3 shows the order of accuracy for the second order Pade’ finite difference scheme at the boundaries and at x = Lx/2 [17]. The compact finite difference scheme is an implicit scheme; hence the highest order of accuracy can be obtained at the maximum distance from both boundaries where the lower order schemes are used. An algebraic mapping given by y = (Ly y0 h)/(y0 + Ly (1 – h))

(30)

is used to map the physical domain y (y Î[0, Ly]) into the computational domain of h (h Î [0, 1]). y0 in Eq. (30) is a stretching parameter of the mapping. Figure 4 compares the physical domain, with the computational domain. The grid spacing in the computational domain are equally spaced, thus we can directly apply the compact finite difference scheme of Lele [16] to compute the derivatives in the computational domain. However, we must use the chain rule of differentiation to find the derivative in y. Application of the chain rule for the first, second and fourth derivative results are \

d d = ηy η d dy

\

d2 d d2 + η2y 2 = ηyy 2 dη dy dη

\

4 d4 d d3 d2 2 2 4 d y = η η η η η η η + + + + 6 4 3 yyyy y y y yyy yy y dη dy 4 dη3 dy 2 dη4

(31)

e

j

The accuracy of the numerical code used to calculate the first and second derivatives of f (y) = exp (– g y), where g , is checked against the exact derivatives. The results, shown in Figures 5 and 6, indicate excellent approximations for the cross-stream derivatives. 1 0.5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

h 1 0.5 0

0

1

2

3 Ly

4

5

6

Fig. 4 Comparison between physical domain (top) and computational domain (down).

A. Zarghami, M. Feizabadi Farahani et. al.

Fig. 5

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First derivative approximation in cross-stream direction for f (y) = 1 – exp (– y).

Fig. 6 Second derivative approximation in cross-stream direction for f (y) = 1 – exp (–y).

4.1 Integration of the Continuity Equation Equation (14) is the governing equation for the cross-stream velocity. Using the compact finite difference scheme, indicate that the right-hand side (RHS) of Eq. (23) experience the ill-conditioning problem. In other words, the diagonal elements of RHS matrix of Eq. (23) are zero. To overcome illconditioning problem, the y derivative operator is applied both sides of Eq. (14).

∂2u ∂2 v = − ∂x∂y ∂y 2

(32)

Equation (32) is not ill-conditioned. This also satisfies the boundary conditions at infinities. In other words, Eq. (32), which is second order differential equation, is solved with v (y = 0) = 0 and v (y = +¥) = 0 as boundary conditions.

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4.2 Solution of the Poisson Equation The application of the time advancement scheme to Eq. (13) generates a Poisson equation for D u (x, y). Ñ2Du =

∂2 u ∂2 u + =C ∂x 2 ∂ y 2

(33)

where C is the linear combination of the RHS of (13). Substituting the second derivatives operators in x and y directions gives (D2X . D uT)T + D2Y. D u = C

(34)

Equation (35) can be written in the following form. (D u. D2X T)T D2Y. D u = C

(35)

Thus, the direct solution of Eq. (33), using a Pade’ finite difference scheme for the second derivatives in x and y, forms a matrix equation of the form AZ + ZB = C. Bartlez [18] has solved the matrix equation AZ + ZB = C. Note that here A = D2Y, B = D2XT and Z = D u.

4.3 Time Discretization A compact, third order, Runge-Kutta time differencing scheme developed by Wray [19] is used to advance the computations in time. Application of the time advancement scheme to the following model equation

∂u = R(u) (36) ∂t is performed in three sub-steps according to Table 1. The table shows that the time advancement of Eq. (36) by one time increment (Dt) requires computation of the right-hand side (R) in three successive sub-time-step. In each of these sub-steps, time (t) is incremented by (ci + di ) Dt and u is accumulated by linear combination of (R) associated with the current time level and that of the previous sub-time-step. Results in the second column of the third sub-time-step are regarded as the solution of the model equation at next time step. In other words, it is the solution incremented by (Dt). The coefficient used in the time advancement scheme (ci , di ) can be obtained using the Taylor series for R¢ and R¢¢ and equating the terms of like orders. This leads to: c1 – c2 – c3 – d1 – d2 – d3 = 1, c12c2 + c3(c1 + c2 ( 1 + c1c2 + c3 (

d2 2 ) ) + c 21d3 = 1/3, c2

d2 d d (1 + 3 ) + c 2 (1 + 2 )) = 1/2 c2 c3 c2 c1c2c3 = 1/6

(37)

A. Zarghami, M. Feizabadi Farahani et. al.

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There are two parameter families of solutions to the preceding set of equations. The scheme will be self-starting if d1 = 0. One parameter families of solution to the set of equations is: c1 = 8/15

d 1 = 0,

c2 = 5/12,

d 2 = –17/60,

d 3 = – 5/12. c3 = 3/4, A test case is performed to verify the order of accuracy for the time-advancement scheme. The equation du/dt = –u (t)

(38)

–t

has an exact solution of u (t) = e when the initial condition is set to u (0) = 1. Hence, the right-hand side of Eq. (38) and the initial condition u (0) = 1 are used to solve for u (t) at t = 1 using different time increment. The maximum errors between the numerical results and the exact solution are shown in Figure 7, which clearly indicates that the order of accuracy is approximately three [17]. T=>lA 1

Third order Runge-Kutta time advancement scheme [19]. Time

1st location

n

u u¢ = un + c1 D tR u¢¢ = u¢ – (c 2 R¢ – d 2R) Dt u n + 1 = u¢¢ + (c3 R¢¢ + d3R¢)D t

Maximum of [Error]

t t¢ = t n + c1Dt t ¢¢ = t¢ – (c2 – d2) D t t n + 1 = t n – Dt

Fig. 7

2nd location

n

R(un) R¢= R(u¢) R¢¢ = R(u¢¢)

n = 3.027253

Order of time advancement scheme for du/dt – u(t) with u(0) = 1.

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International Journal of Mathematical Modeling, Simulation and Applications

5. RESULTS In the absence of external forcing, the results of the simulation display essentially laminar growth at the Reynolds number described above. Figure 8 compares the result of DNS with the Blasious solution. The numerical results show a very good accuracy and agreement with exact solution of the Navier-Stokes equation. 1

0 .8 D N S B .L . B la s iu s B .L .

u/U

0 .6

0 .4

0 .2

2

Fig. 8

4

6

E th a

8

Comparison between DNS results with the Blasious solution.

Figures 9 and 10 illustrate time traces of the results for the velocity components at selected location in the layer. The mean field statistics for the streamwise velocity component and vorticity are illustrated in Figures 11 and 12. Clearly, these results are representative of a self-similar layer.

0.006 X = 0.1 m X = 0.3 m

0.005

X = 0.5 m X = 0.7 m

u

0.004

0.003

0.002

0.001 0

Fig. 9

200

400

Time

600

800

Velocity time histories for u component at streamwise location x = 0.1, 0.3, 0.5 and 0.7.

A. Zarghami, M. Feizabadi Farahani et. al.

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X = 0.1 m X = 0.3 m

0.0002

X = 0.5 m X = 0.7 m

V

0.00015

0.0001

5E-05

0

0

200

400

Time

600

800

Fig. 10 Velocity time histories for v component at streamwise location x = 0.1, 0.3, 0.5 and 0.7. 1 1 0.9

0.8

X = 1.5 m X = 1.3 m

0.8

X = 1.1 m

u/U

0.6

Vorticity

0.7 X = 1.5 m X = 1.3 m

0.5

X = 1.1 m

0.4

0.6

X = 0.9 m

0.4

X = 0.9 m

0.3

0.2

0.2 0.1 2

Fig. 11

4

Etha

6

8

2

10

Mean field statistics for U/U¥.

Fig. 12

4

Etha

6

8

10

Mean field statistics for (w ´ d)/U¥.

5.1 Forced Boundary Layer The case of forcing considered in this investigation applies a set of time dependant perturbations to the v velocity component at the inlet plane. These perturbations are generated by the means of linear stability analysis for the most unstable mode. The inflow perturbation has strong influence on the growth of the boundary layer. Figures 13 through 16 illustrate time traces of the results for the velocity components at selected location in the layer. The figures clearly indicate that the response of the layer is very periodic. This is due to the periodic forcing imposed at the inlet plane of the layer. The peak-to-peak time lapse in these curves provides evidence of the passage of a structure. This time laps, Dt, together with an assumed advection speed for these structures of U , allows estimation of the scale of a structure.

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The mean field statistics for u velocity components and vorticity are illustrated in Figures 17 and 18. Clearly, these results are not representative of a self-similar layer. This is apparently as a result of the forcing imposed at the inlet plane. As the flow goes downstream, the distributions become closer together. This is indicating that the flow enters the self-similar region.

Fig. 13

Velocity time histories for the u component at streamwise location x = 0.1 m.

Fig. 14

Velocity time histories for the v component at streamwise location x = 0.1 m.

A. Zarghami, M. Feizabadi Farahani et. al.

Fig. 15

Velocity time histories for the u component at streamwise location x = 1.3 m.

Fig. 16

Velocity time histories for the v component at streamwise location x = 1.3 m.

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International Journal of Mathematical Modeling, Simulation and Applications

Fig. 17

Mean field statistics for U/U¥.

x x x x x x x x

Fig. 18

Mean field statistics for (w ´ d)/U¥.

The turbulence intensities realized for the velocity components in this simulation are presented in Figures 19 and 20. Again, these statistical quantities indicate that the layer is not self-similar. Figures show that the stresses have maximum value near the wall and damp in far from the wall. The curve representing the turbulence intensity of the u2 component (Figure 19) are very different in shape from those of self-similar layer. The peak intensity for this component exceeds that of the v2 component. It is because of damping of the wall normal component, v2 by the presence of a solid boundary. The Reynolds stress statistics obtained from this simulation are illustrated in the Figure 21. Again, these profiles do not exhibit self-similar behavior. The distributions are more likely collapse on each other at far downstream of the flow.

A. Zarghami, M. Feizabadi Farahani et. al.

x= x= x= x= x= x= x=

Fig. 19

Turbulence intensity for u¢/U2.

x= x= x= x= x= x= x=

Fig. 20

Turbulence intensity for n¢2 /U2 .

x= x= x= x= x= x= x=

Fig. 21 Reynolds stress for u′ v ′/U2 .

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International Journal of Mathematical Modeling, Simulation and Applications

6. CONCLUSION The two-dimensional incompressible, spatially developing, forced, plane boundary layer has been simulated in this work. A numerical method which employs a combination of compact finite difference and a mapped compact finite difference scheme are used to represent the spatial dependence of the boundary layer flow. The governing equations are discretized in streamwise and cross-stream direction using a sixth order compact finite difference scheme and a mapped compact finite difference method, respectively. The simulations were time advanced by means of a third order Runge-Kutta method. The inflow boundaries are excited to generate the forced boundary layer solutions. An advection type outflow boundary condition was employed in this work. This condition appears to allow a simulation that does not distort the structures as they exit the computational domain. This simulation reflects the imposition of a time dependant perturbation function at the inlet plane. This perturbation corresponds to the fundamental mode corresponding to solutions to the Orr-Sommerfeld equation for the hyperbolic tangent inviscid shear layer profile. The results of the simulation capture the physics of the forced boundary layer quite well.

REFERENCES 1. Schlichting, H. “Boundary Layer Theory”, 3rd ed. Springer-Verlag, 2005. 2. Rai, M.M., and Moin, P. “Direct Simulation of Turbulent Flow Using Finite Difference Schemes”, J. Comp.Phys., 1991, 96: pp. 15-33. 3. Yang, W.B., Zhang, H.Q., Chan, C.K., and Lin, W.Y. “Large Eddy Simulation of Mixing Layer”, J. Computational and Applied Mathematics, 2004, 163: pp. 311-318. 4. Tenaud, C., Pellerin, S., Dulieu, A., and Phuoc, L. “Large Eddy Simulation of a Spatially Developing Incompressible 3D Mixing Layer Using the v Formulation”, J. Computers & Fluids, 2005, 34: pp. 6796. 5. Sagaut, P. “Large Eddy Simulation of Incompressible Flows. An Introduction”, Springer-Verlag, 2001. 6. Moin, P., and Mahesh, K, “Direct Numerical Simulation: A Tool in Turbulence Research”, Annu. Rev. Fluid Mech., 1998, 30: pp. 539-578. 7. Hattori, H., Houra, T., and Nagano, Y. “Direct Numerical Simulation of Stable and Unstable Turbulent Thermal Boundary Layers”, Int. J. Heat Fluid Flow, 2007, 28: pp. 1262-1271. 8. Manhart, M., and Friedrich, R. “DNS of a Turbulent Boundary Layer with Separation”, Int. J. Heat Fluid Flow, 2002, 23: pp. 572-578. 9. Hattori, H., and Nagano, Y, “Direct Numerical Simulation of Turbulent Heat Transfer in Plane Impinging Jet”, Int. J. Heat Fluid Flow, 2004, 25: pp. 749-758. 10. Houra, T., and Nagano, Y. “Effects of Adverse Pressure Gradient on Heat Transfer Mechanism in Thermal Boundary Layer”, Int. J. Heat Fluid Flow, 2006, 27: pp. 967-976. 11. Spalart, R.P. “Direct Numerical Simulation of a Turbulent Boundary Layer up to Re q = 1410”, J. Fluid Mech. 1988, 187: pp. 61-98. 12. Lund, T.S., Wu, X., and Squires, K.D. “Generation of Turbulent Inflow Data for SpatiallyDeveloping Boundary Layer Simulation”, J. Comput. Phys. 1998, 140: pp. 233-258. 13. Kong, H., Choi, H., and Lee, J.K. “Direct Numerical Simulation of Turbulent Thermal Boundary Layer”, Phys. Fluids, 2000, 12: pp. 2555-2568.

A. Zarghami, M. Feizabadi Farahani et. al.

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14. Feiereisen, W.J, Reynolds, W.C., and Ferziger, J.H. “Numerical Simulation of a Compressible, Homogeneous Turbulent Shear Flow”, RepTF-13, Thermosci. Dept. Mech. Eng., Stanford, 1981. 15. Li, Q., and Fu, S. “Numerical Simulation of High-Speed Planar Mixing Layer”, J. Computers & Fluids, 2003, 32: pp. 1357-1377. 16. Lele, S.K. “Compact Finite Difference Scheme with Spectral-Like Resolution”, Journal of Computational Physics, 1992, 103: pp. 16-42. 17. Maghrebi, M.J. “A Study of the Evolution of Intense Focal Structures in Spatially-Developing Three-Dimensional Plane Wake”, Ph.D. thesis, Department of Mechanical Engineering, Monash University, Melbourne, Australia, 1999. 18. Bartles, R.H., and Stewart, G.W. “Solution of the Matrix Equation AX + XB = C”, Communications of the ACM, 1972, Vol. 15, Number 9. 19. Wray, A., and Hussaini, M.Y. “Numerical Experiments in Boundary Layer Stability” Proc. R. Soc. Lond, 1984, 392: pp. 373-389.