Transient turbulent flow in a pipe
∗M. S. Ghidaoui† A. A. Kolyshkin‡
R´emi Vaillancourt§
CRM-3176 January 2005
∗This work was supported in part by the Latvian Council of Science, project 04.1239, the Natural Sciences and Engineering Research Council of Canada and the Centre de recherches math´ematiques of the Universit´e de Montr´eal †Hong Kong University of Science and Technology, Department of Civil and Structural engineering, Clear Water Bay, Hong Kong; [email protected]
‡Department of Engineering Mathematics, Riga Technical University, Riga, Latvia LV 1048; [email protected] §Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada;
Abstract
The flow before the rapid deceleration is assumed to be fully developed. Eddy viscosity model is used to describe the flow. The solution is obtained by means of the Laplace transform and the methods of singular perturbation theory. It is shown that the struc-ture of the flow shortly after rapid deceleration is not affected by the eddy viscosity and can be satisfactorily described by a laminar mode.
1999 Mathematics Subject Classification. Primary: 65L06; Secondary: 65D05, 65D30.
Keywords and Phrases. method of matched asymptotic expansions, Laplace transform,
eddy viscosity.
To appear in Scientific Proceedings of Riga Technical University, Computer Science R´sum´e
On mod´elise le ralentissement rapide d’un ´ecoulement pleinement d´evelopp´e au moyen de tourbillons visqueux. On obtient la solution du probl`eme au moyen de la transform´ee de Laplace et de m´ethodes de perturbations singuli`eres. On voit que la viscosit´e des tourbillons n’affecte pas la structure de l’´ecoulement peu de temps apr`es la d´ec´el´eration et que cette structure reste en mode laminaire.
1
Introduction
Recently several turbulence models for the analysis of water hammer flows were proposed in the literature [1]-[4]. These models are used to calculate velocity and pressure distributions for turbulent flows subject to rapid deceleration and/or acceleration. Experiments described in [5] show strong flow asymmetry with respect to the axis of the pipe shortly after a sudden closure of the pipe. Laminar rapidly decelerated flows were studied in [6]-[8]. It is found in [6]-[8] that the velocity profiles after a sudden blockage of the flow contain inflection points and therefore are potentially highly unstable [9]. The analysis in [6], [8] is performed under the assumption that the flow before deceleration is laminar. The case of a turbulent flow is considered in [7]. Approximate velocity profiles after sudden blockage are obtained in [7] under the assumption that the velocity distribution before the deceleration is taken in the form of the ratio of two Bessel functions. The Bessel functions are used in [7] to fit the velocity distribution for a particular range of Reynolds number.
It is shown in [8] that the flow structure after sudden blockage of a fully developed laminar flow in a pipe can be conveniently described by the method of matched asymptotic expansions. Similar solutions were found in [10] and [11] for flows between two parallel planes and in an annulus. In applications the flow before deceleration is usually turbulent. An approach based on an eddy viscosity concept is often used in turbulent flow modeling. Two-layer and five-layer eddy viscosity models are used in [4]. It is shown in [12] that in many cases one can approximate the eddy viscosity distribution in a pipe by a constant in the core region and by a linear function near the wall. This eddy viscosity model is used in the present paper together with the method of matched asymptotic expansions to construct an asymptotic solution for the velocity distribution valid for a short time after the flow is suddenly decelerated.
2
Asymptotic solution
Consider an infinitely long horizontal pipe with radius R filled with a viscous incompressible fluid. At time ˜t = 0 the flow is instantaneously decelerated so that the total fluid flux through the cross-section of the pipe is equal to zero for all ˜t ≥ 0. It is assumed that the flow before deceleration is a fully developed steady turbulent flow which depends on the radial coordinate only. We assume that the velocity vector has only one non-zero component, ˜U(˜r, ˜t), depending on the radial coordinate
and time. The velocity distribution in this case is given by the following equation
∂ ˜U ∂˜t = − 1 ρ ∂ ˜p ∂ ˜z + 1 ˜ r ∂ ∂˜r r(ν + ν˜ T) ∂ ˜U ∂˜r ! , (1)
where the cylindrical polar coordinates (˜r, θ, ˜z) are chosen so that the origin of the system is on the
axis of the pipe, ˜p is the pressure, ρ is the density of the fluid,ν is the viscosity of the fluid and νT
is the turbulent (eddy) viscosity.
Let the measures of length, time, pressure, and velocity be R, T , ρU0R/T , and U0, where T is
the characteristic time and U0 is the characteristic velocity. Equation (1) then can be written in
the following dimensionless form:
∂U ∂t = ϕ(t) + ε r ∂ ∂r rν(r)∂U ∂r , (2) where ε = νT
R2, ϕ(t) = −∂p∂z and the function ν(r) is used to model the distribution of the eddy
viscosity across the pipe. We assume that the eddy viscosity depends only on the radial coordinate,
r.
The boundary conditions are
U(1, t) = 0 and U(0, t) is bounded for all t ≥ 0. (3) The initial condition is
U(r, 0) = g(r), (4)
where g(r) is the function which describes a fully developed steady turbulent flow before deceleration. In addition, the total fluid flux through the cross-section of the pipe is zero for all t ≥ 0, that is,
2
1
Z
0
rU (r, t)dr = 0. (5)
Applying the Laplace transform to (2)-(5) we obtain
s ¯U − g(r) = ¯ϕ + ε r d dr rν(r)dU dr . (6)
The boundary conditions are ¯
U(1, s) = 0 and ¯U(0, s) is bounded. (7) The zero-flux condition is
2
1
Z
0
r ¯U(r, s)dr = 0. (8)
Here ¯U(r, s) is the Laplace transform of U(r, t) and s is the parameter of the Laplace transform.
Experiments [5] and theoretical analysis [7] show that sudden closure of the pipe generates additional vorticity near the wall which starts to diffuse in the radial direction. Thus, the structure of the flow in the core region does not change for a sufficiently short time, but a boundary layer starts to develop near the wall. This physical process suggests that the solution to (6)–(8) can be found by the method of matched asymptotic expansions where the outer part of the expansion corresponds to the core region and the inner part corresponds to the boundary layer near the wall. This method was successfully used in [8], [10], [11] for the case of a laminar flow. Here the method is used for turbulent flows. We assume that the eddy viscosity can be approximated by a linear function of the radial coordinate near the wall and by a constant in the core region as suggested in [12].
We use the following boundary layer variable
ξ = 1 − r√ ε .
The outer expansion (in the core region of the pipe) is sought in the form ¯ U(r, s, ε) = ¯U0(r, s) + √ ε ¯U1(r, s) + ε ¯U2(r, s) + . . . , (9) ¯ ϕ(s, ε) = ¯ϕ0(s) + √ ε ¯ϕ1(s) + ε ¯ϕ2(s) + . . . . (10)
Substituting (9)-(10) into (6) and collecting the terms that do not contain ε, we obtain
s ¯U0 − g(r) = ¯ϕ0, (11)
2
1
Z
0
r ¯U0(r, s)dr = 0. (12)
It can easily be shown from (11) and (12) that ¯ U0(r, s) = g(r) s − G s, ϕ¯0(s) = −G, (13)
where G is the average velocity of undisturbed flow:
G = 2
1
Z
0
rg(r)dr. (14)
The inner expansion (near the wall r = 1) is sought in the form ¯
U(r, s, ε) = ¯u0(ξ, s) +
√
ε¯u1(ξ, s) + ε¯u2(ξ, s) + . . . , (15)
Substituting (15) into (6) and collecting the terms that do not containε , we obtain
d2u¯ 0
dξ2 − s¯u0 = G. (16)
The general solution to (16) is ¯ u0 = C1exp √ s, ξ+ C2exp − √ s ξ− G s. (17)
Using the zero boundary condition at ξ = 0 and the matching condition lim
ξ→∞u¯0(ξ, s) = limr→1
¯
U0(r, s)
we obtain C1 = 0 and C2 = G/s. Thus, the function ¯u0 is
¯ u0 = G s exp − √ s ξ− G s (18)
and a uniformly valid approximation (0 ≤ r ≤ 1) of order unity is ¯ U(r, s, ε) = g(r) s − G s + G s exp − √ s ξ+ O √ε. (19)
It is interesting to note that the solution to the leading order (19) is independent of the eddy viscosity ν(r) and has the same structure as the solution for the laminar case [8]. This means, in particular, that for a very short time there is no change in the structure of the turbulence (that is, the turbulence is “frozen”). This fact is consistent with the numerical experiments [4] where it is shown that for short time it does not matter which eddy viscosity model is used to describe the turbulence — all models give essentially the same dissipation as a laminar flow model.
In order to find the higher terms of the asymptotic expansion we substitute (19) into (8) and use (9). This gives the condition
2 1 Z 0 r ¯U1dr = −2 G √ s. (20) 3
The equation for the functions ¯U1 and ¯ϕ1 in the core region of the flow is
s ¯U1 = ¯ϕ1. (21)
It follows from (20) and (21) that ¯ U1 = −2 G s√s, ϕ¯1 = −2 G √ s.
Following [12] we assume that the eddy viscosity ν(r) is constant in the core region of the flow and is a linear function of r near the wall r = 1, that is,
ν(r) = 1 + α(1 − r) (22)
in the boundary layer, where α is a constant. Substituting (15) into (6) and (7), using (22) and collecting the terms of order√ε, we obtain
d2u¯ 1 dξ2 − s¯u1 = − ¯ϕ1− (α − 1) d¯u0 dξ − αξ d2u¯ 0 dξ2 . (23)
The boundary condition is
¯
u1(0, s) = 0. (24)
Solving (23) and (24), using (18) and the matching condition lim
ξ→∞u¯1(ξ, s) = limr→1
¯
U1(r, s),
we obtain the function ¯u1(ξ, s) in the form
¯ u1(ξ, s) = 2G s√s exp −√s ξ − 1+ αG 4√sξ 2exp −√s ξ+ G 2s − αG 4s ξ exp −√s ξ. (25) Thus, the Laplace transform of the solution up to O(ε) is
¯ U(r, s, ε) = g(r) s − G s + G s exp − √ s ξ +√ε 2G s√s exp − √ s ξ− 2G s√s + αG 4√sξ 2exp −√s ξ + G 2s − αG 4s ξ exp −√s ξ + O(ε), (26) where ξ = 1−r√ ε.
Using the formulas
L−1 exp (−β√s) s = erfc β 2√t , L−1 exp (−β√s) √ s = √1 πtexp −β 2 4t and L−1 exp (−β√s) s√s = 2 r t π exp −β2 4t − β erfc β 2√t ,
where L−1[F (s)] = f (t) is the inverse Laplace transform of the function F (s), we obtain the inverse
Laplace transform of (26) in the form
U(r, t) = g(r) − G + (3r − 1)G 2 erfc 1 − r 2√εt + 4G r εt π exp −(1 − r) 2 4εt − 1 + αG 4√πεt(1 − r) 2exp −(1 − r) 2 4εt + O(ε). 4
3
Conclusion
A transient rapidly decelerated pipe flow is considered in the present paper. The flow before deceleration is assumed to be fully developed. At t = 0 the flow is instantaneously decelerated so that the total fluid flux through the cross-section of the pipe is equal to zero. The method of matched asymptotic expansions is used to construct the solution.
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