0093-6413/00IS-see front matter
Plh S0093-6413(00)00102-6
L I N E A R I N S T A B I L I T Y IN T H E N E A R W A K E
O F A S Y M M E T R I C A L J U N C T I O N O F T W O C H A N N E L S
A. ACHIQ
U. F. R. de Mecanique - Facult6 des Sciences et Techniques BP 146 Mohammadia Morocco
J. KHALID NACIRt
U. F. R. de Mrcanique - Facult6 des Sciences Ain Chock BP 5366 M a ~ i f Casablanca, Morocco
(Received 13 September 1999; accepted for print 30 January 2000)
Abstract - The growth of linear disturbances in the high-Reynolds number laminar wake of a symmetrical junction of two channels is investigated. Disturbances in the near wake respond according to the Rayleigh equation which is solved numerically for a spatial stability problem.
The unperturbed flow is studied in the same way as Goldstein's analysis. The results of this analysis allow for the basic flow profiles at different streamwise positions to be numerically obtained as solutions of the wake boundary-layer equations.
Introduction
Branching and merging flows are of especial interest in practical terms in internal machinery dynamics, and they are also of especial interest more generally with regard to the understanding of fundamental fluid dynamics. The present work is concerned with the flow of an incompressible fluid through a divided channel. The flow involved is the wake kind in the sense that it forces two oncoming channel flows to join into one. Many investigations, experimental, numerical, and analytical, have been made of the external flow past a trailing, the main application being to aerodynamics (for example : the aerofoil wakes problem [1]). However, the properties of the flow past a trailing edge within a channel have received scant attention.
The specific geometry consists of an infinitely long straight channel I Y*l--~, -oo<X*< o0 of width 2L containing a symmetrically disposed straight splitter plate of semi-infinite extent given by Y*=O, X*< 0 (fig. 1). Here Of* Y*) are Cartesian coordinates with origin at the trailing edge of the splitter plate, while the corresponding velocity component (U*,V*) satisfies U*--,'QIY*I(L- I Y*DL "3, V*---~O as far upstream as X*--~oo. These conditions reflect the presence of oncoming plane Poiseuille flow in the two channels far upstream, the mass flux in each channel being Q>0.
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Far downstream the plane Poiseuille flow appropriate to the single wider channel is supposed to emerge, so that U*---)Q(L2-y*2)/4L 3, V*---~O as X*--9~. In between these two limiting flows, the motion is assumed to be steady, laminar and two-dimensional.
;y*
L*!
~ \ ) Upstream Near Far \~
interaction : .__ _w_ak_ e_ . . . w_ _ake_ .. . . ~___
• 0 ,' X*
/
~ j / ' c j"
-L * Figure 1.
The global study of this problem involve three regions ; the upstream interactions region, the near wake just downstream of the trailing edge and the far wake (fig. 1). Using a theory based on Smith's free-interaction structure [2], Badr & coll. [3] studied the upstream interactions problem and its asymptotic matching with the near wake. In this paper, we choose to focus on the near wake problem and its stability. Then small perturbations in the flow quantities are introduced in the wake beyond the trailing edge, but still outside the free-interactive structure region.
This work is constituted of two main parts, the first (§I), dedicated to the asymptotic and numeric treatment of the steady basic flow, allows us to get the unperturbed velocity profiles at different streamwise positions just downstream of the trailing edge. In the next part (§11), using the quasi- parallel form of the basic flow for high Reynolds numbers, the linear stability problem is formulated and the numerical method used to solve the Rayleigh equation is described. Finally, the numerical results of the spatial stability problem are presented and commented upon.
I. THE STEADY BASIC P R O B L E M
1.1. Problem formulation
The fluid is assumed to be Newtonian and incompressible, and its motion to be laminar and steady, To nondimensionalize the two-dimensional problem, we use coordinates X, Y corresponding velocity components U, V and the pressure P, nondimensionalised with respect to L, L, U , U and r/UL -~ respectively, where O- = 6Q/L, and r/is the viscosity of the fluid. The density p is assumed to be constant, the dimensionless governing equations can be written :
Re(V. V) V : - V P + A V and V. f : 0 (1)
where V = (0/0X, 8/c7Y), A = 8 2 / 8 X 2 + 0 2/~'iz2, V ~ (U, V) and Re =p L U / r I the Reynolds number
supposed to be large (ReM). The symmetry of the problem in I1=0 reduces the domain of study
to O_<Y_<I for the Y coordinate, and the Poiseuille flow in the channel far upstream as X---~-oo is
given, in dimensionless form, by U(X,Y)=Uo(Y)=Y-Y 2. It should be noted that the convective
acceleration term in equation (1), which is equal to zero for the Poiseuille flow, can no longer be
negligible because of the great magnitude o f Re. Then, noting that the far downstream flow has a different nature, it is necessary to introduce an intermediate region where convective terms balance viscous terms. In this zone, we have the following scales : X=O(Re), U=O(1), V=O(Re9
and P=O(Re) [3,4]. Therefore, we introduce x=X/Re, y=Y, u=U, v=VRe and p=P/Re, then neglecting the O(Re -l) term the fluid flow is governed by the classical boundary layer equations :
u 3 u / & + v ~ / 3 y = - d p / d r + ~ ~ u / ~ 2 ; ~ / ~ + 3v/3y=O (2) In the wake (x>0) the boundary conditions at y=0 and y=l are given by:
v(y = O)=cgu/dy(y=O)-O ; u(y = 1) = v(y = 1 ) - 0 (3) and the initial conditions at x=0 + are set by the Poiseuille distribution :
u(x=O+,y) : Uo(y) : y - y 2 ; v(x:O+,y):O (4)
1.2. Near wake analysis
The nature of the singularity at x=O leads us to adopt for the near wake (O¢Xul) an analogous analysis to the one used by Goldstein [5]. The stream function ~/defined by : u=o3g/@ and v=-O~,/cOx allows the system (2,3,4) can be written a s
~ V 02~ O~u 02V dp ~3 V
- - - + - - ( 5 )
~ y & ~ y ~ 0 y 2 a~ ~y3 with the boundary conditions :
~,(r,y = 0) -- ~ ' Oy2 ( x y = O) = ' ( x , y = 1) = 0 and q / ( x , y = 1) = 1/6 (6)
The Poiseuille distribution allows in first approximation Vo(y)=y2/2-j/3, which does not verify the symmetry condition at y=0. Therefore we consider a boundary layer for 0, y ul.
1.2.a. Lower boundary layer (y~O)
We introduce transformations ~=x m (x<<l), rl-y/3~ (rl=O(1)), V(x,y)=ff(~,rl) and the following developments : f(~, rl)=fo(r l) + ~ f l ( r l) + ... and p(x) - p o ÷ ~ p l + ¢ p2 ~ ..., where p , and f,(rl) are respectively constants and functions to be determined. The equation (5) and boundary conditions (6) permit to get p l - 0 and the system :
f'o" + 2 fof~' - f'o 2 = 18p2 ; fo(O) : f~'(O) = 0 ; f~(rl) - 9r I ---> 6 when 77 --~ ac (7)
where 6 is a unknown constant related to the displacement of the main flow [6]. Note that ~ will
be determined later by a matching condition. At this stage, system (7) can not be resolved
because of the presence of two unknown constant 8andp2. Therefore, we develop the solution of equation (5) in the main flow for 0<):<1.
1.2.b. Main flow (O<y<l)
While considering the following development ~,(x,y)=~Uo(y)+~l(y)+~e~u2(y)+ .... where ~'n(y) are functions to be determined. Equation (5) and the asymptotic matching with (7) permits :
~u(x,y) : ( J / 2 - j / 3 ) + ~ ~ (y-i)+ O ( f ) (8) This result does not verify the no slip condition at y 1. Therefore we introduce an upper boundary layer for 0<1-y<<1.
1.2.c. Upper boundary layer (y~l)
Near the upper boundary y - l , we introduce transformations ~ - H - y ) / 3 ~ ( ~ - 0 ( 1 ) ) and tu(x,y) = ~ f (~, ~ ) , and we consider the following development: f (~, ~) : fo ( ~ ) + ~ f~ ( ~ ) + ...
where Yn(0)are functions to be determined. Equation (5) and boundary conditions (6) permit obtention of the system :
fo + 2fofo - fo : 18p2 ; fo(O) = fo'(O) : 0 ; fo ( ~ ) - 9~ --~ - 6 when ~ ---> ~ (9) where the last condition in (9) insure the asymptotic matching between the upper boundary layer (yM) and the main flow (O<y< 1) described by (7).
1.3. Results and discussion 1; y
The numerical resolution of the interlinked similarity problems (7) and (9) was obtained simultaneously. A shooting technique was employed to obtain the solution satisfying the identical constants 8 and p2 in each system. This resolution permits obtention of the values 8~1. 62, p2~O. 32 andfo '(0)~3.95, and deduction of the pressure gradient and the velocity at y - 0 :
Figure 2 : Unperturbed velocity profiles u(y) at different positions in the near wake
(~ = 0.00, 0.02, 0.04, 0.06, 0.08, o. 1, o. 12, o. 14, 0.16, 0.18, 0.2)
dp = 2 _ x-J/3 +... ~ 0.21x-J/3 . u(x,y= O) ~. lx;/3 fo'(O ) ~ 1.32x m
dx 3P2 3 (10)
To obtain the unperturbed velocity profiles between y=O and y=l, we choose to resolve numerically system (2,3,4) completed by conditions (10). Some of the calculations of the streamwise component of the flow U(X, Y)=u(y), at increasing downstream stations, are shown in figure 2. The streamwise coordinate indicated is ~ and is given by ~=XVSRe :Is, where X is the non-dimensional distance from the trailing edge.
The transition from the double Poiseuille flow at x=O to a far wake flow is shown in figure 2. So following (8), we have an acceleration of the flow at O_<y< 1/2, caused by the slipping of the fluid aider the trailing edge of the plate. For 1/2<)_<1, the flow is decelerated in order to maintain a constant flow rate across section (O_<y_<l).
IL STUDY OF THE LINEAR STABILITY PROBLEM ILl. Formulation of linear stability
The purpose of this part is to study the linear stability of the near wake. Formally, small perturbations in the flow quantities are introduced in the wake beyond the trailing edge. From the previous analysis, we can deduce the following scales : O(V)=O(oW/OX)=O(Rem). At high Reynolds numbers, the basic flow at a fixed X-station can be written : U(X,Y)=U(Y)=u(Y) and
VO(,Y)=O. To study the stability of the flow we let :
U(X,Y,O=U(Y)+U (X,Y,O ; V(X,Y,O=IV (X,Y,O and P(X,Y,O=P(X)+ P (X,Y,O, where (U, IV) is the velocity disturbance and P is the pressure disturbance. By substituting these expressions in equations (1) and neglecting the quadratic term in the velocity disturbance [7], we obtain the linearized and inviscid equations of motion :
a +cOO+vOQ= ; a +caiv ; - - + - - oO = o (! 1)
at OX aY aX at ax aY ax oY
The Rayleigh equation is derived by the introduction of the stream function and normal-mode analysis. More specifically, the stream function is taken to be ~ (X,y,O=qJ(y) exp[i(~-ca 0], where a and ca are respectively wavenumber and frequency and they are complex i.e. a Ctr+ia, and ca=o~+io~. Substitution into the system (1) gives the classical Rayleigh stability equation :
( a U-<o) ( # " - a e q,) = a U " q, (12)
which, together with the boundary conditions
q~(O)=qa(l):O (13)
defines the basic eigenvalue problem for this inviscid quasi-parallel shear flows. In fact
Rayleigh's stability equation is the vorticity equation of the disturbance.
11.2. Results and discussion
For the stability of the wake, it has been observed both experimentally and'numerically [8] that thespatial growth rates describe the stability characteristics very well. So we choose to focus on spatial stability problem (a~=0). In this case the disturbances are characterised by a real frequency co, a spatial growth rate ai and a wavelength 2=2~ar. For numerical resolution of (12,13) we use the method described by Papageorgiou & coil [9] who considered the linear instability of the wake behind a fiat plate placed parallel to a uniform stream.
Figure 3.
Spatial stability of proper wake profiles at various streamwise positions. Curves of neutral stability in (~, co), from left to right,
at : ~= 0.02, 0.04, 0.06, 0.08 and 0.1.
0
