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0Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 1, pp. 113-125. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd240202
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 76A05, 76D10, 76M45
Prandtl System of Equations with Self-Induced Pressure for the Case of Non-Newtonian Fluid: Dynamics of Boundary Layer Separation
R. K. Gaydukov
The problem of flow of a non-Newtonian viscous fluid with power-law rheological properties along a semi-infinite plate with a small localized irregularity on the surface is considered for large Reynolds numbers. The asymptotic solution with double-deck structure of the boundary layer is constructed. The numerical simulation of the flow in the region near the surface was performed for different fluid indices. The results of investigations of the flow properties depending on the fluid index are presented. Namely, the boundary layer separation is investigated for different fluid indices, and the dynamics of vortex formation in this region is shown.
Keywords: double-deck structure, boundary layer separation, power-law fluid, localized perturbations, asymptotics, numerical simulation
1. Introduction
The problems of a viscous fluid laminar flow along slightly rough surfaces for large Reynolds numbers have been studied for a long time. One of the significant properties of flows in such problems is the existence of boundary layer separation, i.e., of regions near the surface with vortices inside, due to which such problems cannot be simply described by the classical Prandtl boundary layer theory [1]. An efficient approach to modeling the flow in problems of this type has been known since the 1960s — the multi-deck (double- and triple-deck) boundary layer theory. The triple-deck model was first obtained by V. Ya. Neiland [2], K. Stewartson and P.G.Williams [3], and was first applied to problems of flow along a small hump by F. T. Smith [4]. The double-deck
Received November 16, 2023 Accepted December 25, 2023
The work was supported by the Russian Science Foundation under grant No. 19-71-10003 (https://rscf.ru/en/project/19-71-10003/).
Roman K. Gaydukov roma1990@gmail.com
HSE University
ul. Myasnitskaya 20, Moscow, 101000 Russia
structure discovered a little later by J. Mauss [5] and V. G. Danilov [6] appears when considering geometric scales other than for the triple-deck structure, see [7] for details.
In this theory, using asymptotic methods, the system of Navier-Stokes and continuity equations is reduced to a series of simpler systems describing the flow in the "decks", i.e., in boundary layers with different scales. These reduced problems do not contain spatial multi-scales that were present in the original system due to the geometric scales of irregularities, and they can be efficiently solved numerically by using simple difference schemes. One of such problems is the Prandtl system of equations with self-induced pressure [4, 8, 9] describing the flow near the streamlined surface d = h(£) (i.e., in the lower deck, see Fig. 1 and (1.6)), which has the following form for the Newtonian fluid:
du du du dp
dt+ud(+vdd=~d(
du du dv ,
I I n ^U
u\e=h(0 =v\e=h(0 =0> qq
= C, = vU±x = °, u\t=o = uo(£, e),
where u and v are horizontal and vertical velocity components, £ and d are horizontal and vertical boundary layer variables corresponding to geometric scales (width and height) of an irregularity, p is pressure, n is the classical boundary layer variable, and C is a constant depending on the general geometry (i.e., plate, pipe, channel, etc.), see for details below. The difference between
the double- and triple-decks consists in expression for the self-induced pressure , see [7, 10]
^ n=o
for details.
System (1.1) was obtained for Newtonian fluid flow problems in various geometries with a small localized irregularity on the streamlined surface: on a plate [4, 8, 11-14], inside a channel and an axially-symmetric pipe [15-17], and on a uniformly rotating disk [18, 19]. Despite the fact that the multi-deck theory of boundary layer has been widely studied for 60 years [20-26] (see the papers cited above), starting with the pioneering papers by K. Stewartson [27] and F. T. Smith [4], as far as we know, the problems of non-Newtonian fluid flow have not been considered within the framework of this theory.
The goal of this paper is to generalize the multi-deck (in particular, the double-deck) structure to the case of non-Newtonian fluid flow, and to investigate the generalization of system (1.1) to this case of fluid. Of interest is the appearance and dynamics of the boundary layer separation region depending on fluid properties.
Namely, we consider a non-Newtonian fluid with Ostwald - de Waele (power-law) rheological properties [28, 29]. Our goal is to construct the double-deck structure in the case of fluids with this rheology for the flow problem along a semi-infinite plate with small localized irregularities on the surface for large Reynolds numbers Re, see Fig. 1. We note that the double-deck structure was studied for this problem in the case of Newtonian fluid in [8, 11, 13].
As usual, we first introduce the dimensionless variables: the 2^-velocity vector U = (u, v) = = the coordinates (x, y) = the time t = j-, and the pressure p = where the
variables with "hat" are dimensional, the characteristic speed ux is the speed of the upstream plane-parallel flow, the characteristic length x0 is the distance to the irregularity from the edge of the plate, i0 = tt~ is the characteristic time, p^ is the density of the upstream flow, and P = = u2xpx. Thus, the upstream flow is a plane-parallel flow with dimensionless velocity Ux = = (1, 0). Note that x0 ^ S, where S > 0 is some (small) distance from the edge of the plate such that the generalized Prandtl boundary layer has been formed.
nV
EXT
n 0(e)
y* / yw ;o(e4/3) ,
0 x0 - X
' 0(e)
Fig. 1. Localized irregularity on the plate and the double-deck structure: I is the lower deck, II is the main deck, EXT is the region with unperturbed flow
As was written above, we focused on the Ostwald - de Waele power-law model [28] which is described by the stress tensor of the form
r = 2^(7)7,
where the rate-of-strain tensor is 7 = VU + (VU)T, and the apparent (effective) viscosity is
My) = (277)(n-1)/2 =
du dx
+ 2
dv\ ( du dv dy J \dy dx
(n-1)/2
(1.2)
Here n and Jl are real positive parameters (dimensionless constants) called, respectively, the fluid index and consistency. Note that the case n < 1 corresponds to the pseudo-plastic (shear-thinning) fluid, the case n > 1 is known as the dilatant (shear-thickening) fluid, and the case n = 1 corresponds to the Newtonian fluid.
The problem under study is described by the system of Navier-Stokes and continuity equations
2/3 du du du dp
--1-1/.--1- it— = ——
dt ^ 11 dx ^1 dy ,dv
dx
+ é
n+1
2/3 dv dv dv dp n,i
dt dx dy dy
du, dv ^
dx dy
A +2 ^U ^ + ( + ^U ^ ^ dx dx \ dx dy J dy
dy dy \dx dy J dx
(1.3)
(1.4)
(1.5)
where e = Re_1/(ra+1) is a small parameter when the (generalized) Reynolds number Re = = UQ~2Ln!= is large. The coefficient e~'2/3 at. the time derivative appears due to the selected time scale. Namely, in the nonstationary double-deck structure, there arise nontrivial time scales in the layers (or, in the decks, as is more common in this theory) in addition to the well-known spatial scales, see [7] for details.
We assume that the plate surface is described by the formula
= é4/3h
x1
(1.6)
2
2
2
y
s
where the function h(£) is a smooth function such that h\^±x = 0, x = 1 is the point at which the irregularity is localized. Note that the selected geometric scales of irregularity (the powers of the small parameter e in (1.6)) lead to the double-deck structure of the boundary layer, see [7, 8, 11, 13].
Note that, if we consider the streamlined surface in the form ys = ea (Jj), where a > 0, /3 > 0, then the asymptotic solution to the problem will have a different form depending on the values of these parameters. The region near the surface can be described either by the investigated system of Prandtl equations with a different expression for the self-induced pressure, or simply by a linearized system of Prandtl equations (second-order asymptotic corrections); see [14] for a detailed analysis. Note that the qualitative behavior of the solution to the system of Prandtl equations with self-induced pressure in the case of a Newtonian fluid does not depend on the type of expression for pressure, the difference is only quantitative.
System (1.3)-(1.5) is supplemented with the following non-slip condition on the surface ys and the conditions for matching with the upstream (external) flow:
U\ =(0, 0), Ul = Ul >0 = U^. (1.7)
\y=ys v ' n Ix—y—^o ly—x, x>0 x v '
Note that the boundary layer theory for the power-law fluid flow problem along a flat plate (i.e., in the unperturbed case h(£) = 0) has been developed and extensively studied in [30]. The flow is described (see (2.2)) in terms of the solution of the generalized Blasius equation (2.3), which was investigated analytically and numerically, see [31, 32]. We stress that one of the most important results is that the boundary layer has a finite thickness [33].
2. Double-deck structure
We introduce the following boundary layer variables (characteristic scales):
x 1 z ~ z , _ .
£ = —V = e = (2-1)
e e e4'6
where z = y — ys is the new vertical variable such that the lower boundary (the plate surface) becomes flat (y = ys becomes z = 0). The variables 9 and n are the vertical boundary layer variables in the lower deck and in the main deck of the double-deck structure, respectively, see regions I and II in Fig. 1. The variable £ is the horizontal fast variable.
Note that the asymptotic solution of unperturbed (i.e., for the case h(£) = 0) problem (1.2)-(1.7) has the form [30]:
unp{x, y) = l + u{x, r])+0{e), vnp{x, y) = ev{x, rj) +O (t2), pnp{x, y) = p + 0{e), (2.2)
where u(x, rj) = f'n{7) —1, v{x, rj) = (n + l)_1ar1+1/(ra+1)(7/4(7) -/„(7)), P = const, 7 =
and fn(Y) is a generalized Blasius function [31, 32] which is the solution of the boundary-value
problem
(IfT-'fn)' + ^YUn = 0, /„(0) = f!M = 0, U00) = 1. (2.3)
We assume that the initial data for problem (1.3)-(1.7) have the form
= 1 +u(x, r])+e1/3u1^, 9) + 0 (t~2/3), v\t=Q = e2/3 (v1^, 9) + vn(£, r]))+0(e). (2.4)
u\t=0
This choice of initial data means the presence of a steady-state generalized Blasius boundary layer and nonstationary perturbations caused by the irregularity on the plate.
We introduce the following notation: the superscript over a function indicates the layer according to Fig. 1 in which it is defined as the boundary layer function.
Definition 1 (boundary layer function). Let q be a boundary layer type variable (q is 9 or n). We call any smooth function f (..., q) a boundary layer function if f (..., q) decreases by the law |q|_w as q ^ where N £ N is sufficiently large.
The algorithm for constructing an asymptotic solution to problem (1.2)-(1.7) is based on a combination of asymptotic methods: Vishik-Lyusternik boundary layer expansions [34] and the localization principle, see also [7, 9]. The last one is based on the assumption that a small irregularity does not affect the flow far from it, i. e., the solution of perturbed problem (1.2)-(1.7) tends to the solution of unperturbed problem (2.2) as £ ^ ±cx>. The construction procedure is very similar to that in the Newtonian case, see [8, 11, 13] and references therein. We present some key points of this procedure and describe some necessary changes from [8, 11, 13], but we do not show the calculations in detail. Just note that these calculations are simple but very voluminous.
Namely, the difference from the Newtonian case (see [8, 11, 13]) consists in the presence of the function i on the right-hand sides of equations (1.3) and (1.4). We expand the function i in a series in a small parameter e ^ +0, and the function i becomes
f = é1-n (f + é1/3f + ..),
where ii are functions depending on the functions uj and vj which are terms of asymptotic expansions of velocities in a small parameter e, see Theorem 1 below and [8, 11, 13].
Our goal is to obtain the asymptotic solution of the problem under study up to the first-order terms, and only the expression for ¡0 is required for this, see [8, 11, 13]. Simple calculations lead to the following expression of ¡0:
f o =
duO1 du1 'n 1 dn dd
Repeating the procedure for constructing a solution from [8, 11, 13] and taking into account the above comments about differences, we obtain the following result.
Theorem 1. The formal asymptotic solution of problem (1.2)-(1.7), (2.4) has the form
u(t, x, y) = 1 + ufe n) + e1/3 (ui(t, 9) + u\l(£, n)) + O (e2/3), (2.5)
v(t, x, y) = e2/3 (v2(t, 9) + v2(t, n)) + O(e), (2.6)
p(t, x, y) = Po + e2/3p%(t, n) + O(e), (2.7)
where z = y — ys, 9 = rj = f, £ = , and p0 = const. The function t/.J,1 = f{2 — 1> where fn is the generalized Blasius function [31, 32], i. e., the solution of boundary-value problem (2.3), and tt]1 = h [-^j-j
. The functions u[ and vI are determined by the relations
x=l
1 1 1 dn
, vl = v% — vlo\ „ n=0 2 2 2 'n=°
' x=1
where the functions u\ and are the solutions of the initial boundary-value problem for the generalized Prandtl system of equations with self-induced pressure
du\ * ( du\ dh du]
dp
+
n=o
d_
dd
dU
dd
n— 1
Qui
i
dv*2 du\ dh du
0,
U1 ltf=0 = v2 le=0 = 0, ^ 0
dn
n=o
x=1
, v21
du\
duO1
dn
n=o
x=1
a«!1
with some initial data ul\t=0 = u {£, 0) + +
tions. T equation
(2.8)
(2.9)
n=o
x=1
consistent with the boundary condi-
tions. The function v^1 is the solution of the initial boundary-value problem for the Rayleigh-type
,1/3
d_
dt
J < + (t^ + 1) A^t,? - vf2^1
dn2
x=1
0, v2\=o = v21
,
v£| ^ ^ 0,
2 in^œ
(2.10) (2.11)
with some initial data v2I|i=o = fII(£, n) consistent with the boundary conditions. The pressure p2 is determined from the relation [with condition p2I|ç^_œ
/ff^' + KU + i)
dp
d
= ri/3ii d£ " dt
„II ^<0 ~ V2
dn
dn
(2.12)
x=1
The resulting asymptotic solution (2.5)-(2.7) is very similar to that in the Newtonian case [8, 11, 13], the difference consists only in the terms with second-order derivative in problem (2.8). Note that the expression for the induced pressure can be written in terms of v2 (see [7-9, 11, 13]) as follows:
dp2I
duUo
n=o
dT
n=o
x=1
v21
(2.13)
which allows one to solve the lower deck problem (2.8), (2.9) independently of the Rayleigh-type equation (2.10).
t
v2
3. Numerical modeling of the flow near the surface
In this section, we present the results of modeling of the flow in the lower deck, i.e., in the near-surface region. The flow in this region is described by the generalized Prandtl problem with self-induced pressure (2.8), (2.9), (2.13). As was mentioned above, this problem can be solved independently of the Rayleigh-type equation (2.10). Namely, only the value
duoI
dn
o = /n(0)
n=o
x=1
is required, see Theorem 1. The numerical solution of problem (2.3) for the function fn is widely known [31, 32], and the values f"(0) depending on the fluid index n are obtained in [31, see the values of y in Table 1]. For the calculations illustrated in Fig. 2, we have f0'5(0) ~ 0.33123, f1'(0) « 0.33206, f2'(0) « 0.39979.
The numerical modeling of problem (2.8), (2.9), (2.13) was performed using the developed parallel algorithm (with the CUDA technology of parallelization support) based on the first-order explicit difference schemes, see [10, 11]. We do not show this algorithm in detail because of its simplicity. Just note that the constructed algorithm converges and is computationally efficient. The calculations were performed on the spatial domain [—50, 50]^ x [0, 100]0, and we checked that the error caused by this truncation of the initial half-space spatial domain remains small, i.e., smaller than 10-4 for the considered simulation time, see [10]. Note that problem (2.8), (2.9), (2.13) was solved in the variables (£, 9) in which the lower boundary is flat (i.e., the plate surface is 9 = 0), but in all figures below, the results are shown in the variables (£, 9), 9 = 9 + h, in which the lower boundary is curvilinear (i.e., the plate surface is 9 = h).
We assume that the initial data for problem (2.8), (2.9), (2.13) have the form [8, 11]:
*. f fn(0)(9 + 0.2h9), 0 < 9 < 5, u* L_0 = s ,, (3.1)
1 lfn(0)(9 + h), 9 > 5,
which is a simple laminar flow along the irregularity h(£) that satisfies the boundary conditions (2.9). We assume that the shape of the irregularity has the form
h(£) = Ae—2/4, A = const. (3.2)
Note that the flow for this shape of the hump was investigated in the Newtonian case [8, 11, 13], i.e., for the fluid index n = 1, see the middle column "n = 1" in Fig. 2. Namely, it was found that there exists a critical value A* such that a laminar flow is observed for the case A < A*, and a flow with separation of boundary layer, i.e., a region with vortices, for A > A*. Let A = 5, which corresponds to the case A > A*. The goal of the numerical modeling is to investigate the influence of the fluid index n on the vortex formation process (i.e., on the critical amplitude) and its dynamics.
We consider the values n = 0.5 and n = 2, which correspond to the pseudo-plastic (shear-thinning) and dilatant (shear-thickening) fluids, respectively, and we compare the results of numerical modeling for these fluids with the Newtonian case n = 1. The streamlines of the flow are shown in Fig. 2 at some times. The columns in Fig. 2 correspond to different values of fluid indices, and the rows correspond to different times.
For the Newtonian fluid case with fluid index n = 1 (see the middle column in Fig. 2), the dynamics of vortex formation is the following one. At a time after the beginning of the simulation, a vortex is formed on the right wall of the irregularity (see t1), and then this vortex is growing and moves downstream (see t2 and t3). As the first vortex moves away from the hump, a new vortex is formed near the hump (see t4), and this process is repeated (see t5). These vortices decrease when moving downstream (see t6), they are destroyed at some distance from the hump (also see Fig. 6), and a stationary vortex is finally formed near the hump (see t7).
For the shear-thinning fluid case with fluid index n = 0.5 (see the left column in Fig. 2), we at first observe a similar vortex formation (see t3-t6), but after some time, the flow becomes laminar (see t7) and stationary (see for details below).
For the shear-thickening fluid case with fluid index n = 2 (see the right column in Fig. 2), we observe a more intensive vortex formation process (see t3-t6) with vortices of size larger than
flow direction.
10 o
10 -6-4-2 0
6 8 10
-6-4-2 0 2 4 6 8 10
H?-
6 8 10
-4-2 0 2 4 6 8 10 -6-4-2 0 2 4 6 8 10
-6-4-2 0 2 4 6 8 10
lO CS
—4 —2 0 2 4 6 8 10 -6-4-2 0 2 4 6 8 10
lO CO
L O ^^ ^ ^
-4-2 0 2 4 6 8 10 -6-4-2 0 2 4 6 8 10
o
CO
-4-2 0 2 4 6 8 10 -6-4-2 0 2 4 6 8 10
-4-2 0 2 4 6 8 10 12
o o>
-2 0 2 4 6 8 10
-4-2 0 2 4 6 8 10 12 14 16
o
The streamlines axe similar to t7.
The streamlines are similar to i7.
-6-4-2 0 2 4 6 8 10 12 14 16 18 20
Fig. 2. Streamlines of the flow depending on the fluid index n at some times ti
those in the Newtonian fluid case (n = 1). Moreover, we can see that a vortex arises inside a vortex (see t6, t7). But after some time, a stationary vortex is also formed (see t8). To determine the stationarity of the solution, we introduce the function
X(t) = max |ul\t+st - ul\t\,
where 5t is the time step. If after some t* the function x(t) tends to zero for all t > t*, then we obtain a stationary solution u*. For the considered fluid index, this function is shown in Fig. 3. One can see that all the considered flows become stationary at a time after the beginning of the numerical simulation.
x(t)
The vortex dynamics can be illustrated more clearly if we construct the trajectories of the boundaries of the boundary layer separation regions varying in time. For this, we introduce a function Z (t, £) of the form
C(t, 0 = ^ .
dd
0=0
It is well known [35] that, if there exists a point £* such that ( (t, £*) = 0, then the boundary layer is separated at this point, and consequently, a vortex is formed. Moreover, if there exists a pair of points £± (£+ > such that the function ( (t, is equal to zero at these points and does not change sign between them, then the point is the separation point of the boundary layer, and is the point of its adhesion, and there is a vortex in the region between them, see Fig. 4.
r <£+
Fig. 4. Boundary layer separation and the points (schematically)
Thus, from the trajectories of these zeros £±(t) of the function ((t, £) in time, one can easily find the regions of vortex formation. These regions are colored gray in Figs. 5-7. Note that the times ti in these figures are the same as in Fig. 2.
In Fig. 5, we show the vortex regions for n = 0.5. One can see that two vortices are formed (one appears later than the other) and then destroyed after some time, and the flow becomes completely laminar.
In Fig. 6, we show the vortex regions for n = 1. One can see that three vortices are formed (the next vortex is formed by separating from the previous one) and then destroyed after some time. However, the stationary vortex is formed "in the shade" of the hump at time tvs, and after the remaining vortices are destroyed, the flow becomes laminar with a stationary vortex.
In Fig. 7, we show the vortex regions for n = 2. The situation is similar to that in the case n = 1, but the vortex formation process is more intensive: one can see that a vortex is formed inside another vortex (see t6 and Fig. 2). But after some time, the flow becomes laminar with a stationary vortex (the size of the vortex is larger than that in the case n = 1, see Fig. 6).
vortices are colored gray
Fig. 6. Trajectories of zeros of the function Z(t, £) for the flow fluid index n =1, the regions with vortices are colored gray
24
20
22
10
14
16
18
12
4
2
8
6
__11-11 [ i ................................................................. ......
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 151
Fig. 7. Trajectories of zeros of the function Z(t, £) for the flow fluid index n = 2, the regions with vortices are colored gray
4. Conclusions
The asymptotic solution with double-deck structure was constructed for the power-law non-Newtonian fluid flow along a localized irregularity (like a hump) on a semi-infinite plate for large Reynolds numbers, see Theorem 1. The flow near the surface (in the lower deck) was investigated numerically. Namely, the influence of the fluid index on the flow behavior was shown for the same amplitude of irregularities.
The results obtained are as follows. If we decrease the fluid index, then the vortex formation process becomes weaker, and we observe a steady-state laminar flow after some time from the beginning of the numerical modeling. If we increase the fluid index, then the vortex formation process becomes more intensive with vortices of size greater than that in the Newtonian case [8, 11, 13]. But, as in the Newtonian case, we observe the formation of a stationary vortex "in the shade" of the hump after some time from the beginning of the numerical modeling. In fact, this means that the critical amplitude A* (the value of the hump amplitude such that vortices begin to arise in the flow when this value is exceeded) increases with decreasing fluid index n.
Finally, we stress that the results of this paper generalize the theory of double-deck structure of the boundary layer to the case of non-Newtonian fluid with power-law rheology model.
Conflict of interest
The author declares that he has no conflict of interest.
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