Wall shear stress and flow behaviour under transient flow in a pipe
J.M. Abreu
Civil Engineering Dep., Coimbra University, Portugal
A. Betâmio de Almeida
Technical University of Lisbon, CEHIDRO-IST, Portugal
ABSTRACT
A hybrid 2D model is used to analyse the flow behaviour, both in laminar and turbulent
transient flows. Wall shear stress is calculated from the cross-sectional distribution of axial
velocities. Approximate 1-D representations for wall shear stress, τw, is compared with those of
hybrid model and their applicability is examined.
An unsteady generalization of the quasi-stationary relationship between τw, and the head loss J
for unit pipe length, permits to split the additional contribution due to unsteadiness into
instantaneous energy dissipation effects and virtual inertial effects. The relative importance of
these two effects is investigated. An examination of the range of applicability of the
instantaneous-acceleration approach of representing unsteady friction is presented. Lastly, the
relation between the value of the empirical parameter k of the additional acceleration term and
Boussinesq momentum coefficient, β, is clarified.
Although the calculated results of pressure with 1D representation for τw agree well with the
hybrid’s ones and experimental data in a laminar flow, the deviations become significant in
turbulent flow when the time scales of the water hammer and radial diffusion are comparable.
1 INTRODUCTION
The effect of energy dissipation or hydraulic head losses in the simulation of transient fluid pipe
systems has been an important topic of research for a number of years. This effect is especially
relevant to the determination of extreme pressures and to the simulation of compound manoeuvres
for long unsteady flows subject to automatic regulation.
In pressure pipe flows, the head loss essentially depends at any moment on the radial distribution
of axial velocity, whose determination requires a detailed knowledge of the characteristics of the
turbulent flow inside the pipe. For a more precise description of the energy dissipation in
unsteady flows, several researchers have used complex, and thus less limiting, mathematical
models (2D or quasi-2D models). A computational structure, basically consisting of an
axisymmetric model coupled to a 1D model and to a turbulence model, known as a hybrid model,
has also been developed by the authors (Abreu and Almeida, 2000). In terms of practical
engineering applications, this type of approach still cannot compete with the versatility and
calculation time of the 1D models, except in a limited number of situations where the simplicity of
the hydraulic system being studied or the accuracy of the intended results so justifies. But it can be
a valuable tool because it allows systematic tests to be carried out with a view to improving our
understanding of the physical process in detail. In particular, it immediately identifies the need to
characterize and parameterise the wall shear stress, τw, in terms of the characteristics of average
flow, and to incorporate these formulations into 1D models, so as provide realistic simulations to
practical engineering problems. This paper is intended to help towards achieving this goal.
2 MATHEMATICAL FORMULATION OF THE PHYSICAL PROCESS
2.1
The axi-symmetric approximation
Adopting a system of cylindrical coordinates (x, r, θ), and admitting the following simplifying
hypotheses:
• the flow can be regarded as axisymmetric;
• the fluid is held to be homogeneous and monophasic;
• the temperature variations are sufficiently small for the molecular viscosity to be regarded
as constant (isothermal assumption);
the full governing Navier-Stokes equations constitute a system of three equations and four
unknowns (ρ, p, u, v) which is complemented by the equation of state. In keeping with the above
hypotheses, the Navier-Stokes system is reduced to:
• Continuity equation:
∂p
∂p
∂p
 ∂u ∂v v 
+ +u
+v
=0
+ ρ c2  +
∂t
∂x
∂r
 ∂x ∂r r 
•
(1)
Momentum equations:
 ν ∂  ∂u ∂v v 
+
+  + gx
 +

 3 ∂x  ∂x ∂r r 
(2)
2
 ∂2 v
∂v
∂v
∂v
1 ∂p
1 ∂v v  ν ∂  ∂u ∂v v 
∂ v
+u
+v
= −
+ ν  2 +
+
− +
+
+  + gr

2
∂t
∂x
∂r
∂r
ρ ∂r
r ∂r r 2  3 ∂r  ∂x ∂r r 
∂x
(3)
∂u
∂u
∂u
1 ∂p
+u
+v
=−
+ν
∂t
∂x
∂r
ρ ∂x
 ∂ 2 u ∂ 2 u 1 ∂u
+ 2+

2
r ∂r
 ∂ x ∂r
in which u and v represent the axial and radial components of instantaneous velocity; p is the static
pressure; gx and gr are the acceleration of gravity according to x and r, and ρ is the density of a
fluid of kinematic viscosity ν. The second viscosity is considered null (Stokes condition).
Mitra and Rouleau (1983) have solved this system of equations numerically based on the
Warming and Beam factored scheme. They concluded that in the extreme case of a quasiinstantaneous closure, even though the radial pressure variations in the valve section might be
significant, the average pressure in the cross-section is practically identical to that supplied by a
1D model. Upstream from the valve the radial pressure variations decreased quickly and the
pressure wave tends to became plane.
With the aim of reducing the calculation time and eliminating some of the numerical and
computational problems inherent to the above system (Eqs 1-3), almost all engineering
applications allow its simplification. Thus, for uniform lengths of pipe whose cross-section has a
typical dimension that is much smaller than the respective lengths, the hypothesis D/L<<1 is
allowed, which enables the radial velocity, v, to be regarded as negligible in relation to the axial
velocity, u. The Mach number, M, which characterizes the compressibility effects, can be
considered as small in relation to the unit, in liquid flow. Comparing the orders of magnitude
of the various terms according to the simplifying conditions it can be concluded that the
pressure is uniform in the cross-section of the pipe (which is why some authors call the
resulting model quasi-2D). Accordingly, the initial system of three equations can be reduced to
two equations. The numerical resolution of this reduced system has been done by Vardy and
Hwang (1991). They showed that the radial velocities are at least three orders of magnitude
less than the respective axial components. Also disregarding the terms of the order M (acoustic
approximation), the convective pressure terms and the convective velocity terms also
disappear, and we obtain the following equations:
∂p
 ∂u ∂v v 
+ ρ c2  +
+ =0
∂t
 ∂x ∂r r 
∂u 1 ∂p
+
+ g sin θ − ν
∂t ρ ∂x
(4)
 ∂ 2 u 1 ∂u 
 2 +
 =0
r ∂r 
 ∂r
(5)
with g x = − g sin θ , and θ being the angle the pipe’s axis makes with the horizontal. Assuming
that the body forces are negligible, these equations coincide with those given by Brown (1962)
and D'Sousa and Oldenburger (1964). Zielke (1968), derived a formal expression for the wall
shear stress based on equation (5):
τ w (t ) =
8 ρν 
1
 V (t ) +
2
D 
t
∫ W (t − tˆ)
0
∂V ˆ ˆ
(t ) dt 
∂t

(6)
in which W(t) is a weighting function, defined by the author, which operates as “memory”.
In the event of turbulent flow, adopting Reynolds’ procedure, in which the instantaneous
dependent variables in (4)-(5) are decomposed into a mean and a fluctuating term, and then taking
the time average of each equation, yields the mean flow equations, referred as Reynolds-average
equations:
 ∂u ∂ v v 
∂p
+ ρ c 2 
+
+  = 0
∂t
 ∂x ∂r r 
(7)
(
)
∂u 1 ∂ p
ν ∂  ∂u  1 ∂
+
+
+ g sin θ −
r
r u´v´ = 0
∂t ρ ∂x
r ∂r  ∂r  r ∂r
(8)
The mean momentum equation is then complicated by a new unknown term involving velocity
correlations that acts as an additional stress (Reynolds stress), leading to the total stress
τ = ρν
∂u
− ρ u ′ v′
∂r
(9)
System (7)-(8) can only be solved for the average values of the dependent variables when the
Reynolds stress term can be characterized, that is, supplementary constitutive equations have to be
obtained to relate the Reynolds stresses to the mean flow itself before the equations can be solved.
Which is, in effect, the purpose of a turbulence model (Rodi, 1984; Wilcox, 1993).
2.2
The basic equations over a cross-section. One-dimensional model.
Integrating (7) and (8), relative to the cross-section, A, of the pipe, employing boundary condition:
v (r = R ) = ∂Wr / ∂t , in which Wr is the radial displacement of the pipe walls, we get the classical
one-dimensional (1D) equations of the simplified elastic model (Chaudhry, 1987; Wylie and
Streeter, 1993):
∂H a 2 ∂V
+
=0
∂t
g ∂x
(10)
4τ w
∂V
∂H
+g
+
=0
ρD
∂t
∂x
(11)
afterwards, the relation between the pressure and the instantaneous piezometric head, p=ρg(H-z),
in which the fluid density corresponds to determined reference conditions (that is, keeping a
constant value during a transient regime) and z is the elevation of the pipe axis (assumed to be
fixed) at position x. V is the average velocity, in the axial direction, in each section of the pipe.
On integrating the continuity equation, it is assumed, on a first approximation and in accordance
with the classic theory of water hammer, that the Korteweg’s expression for the pressure wave
speed in a elastic pipe, a, may be held to be valid. In accordance with the preceding hypotheses,
the sole effect of the elasticity of the pipe, relative to a perfectly rigid pipe, is to increase the
effective compressibility of the fluid.
When it is valid to consider that the flow could be modelled according to the hypothesis of the
incompressibility of fluid and the indeformability of the pipe (rigid model), the continuity
equation (10) can be disregarded and the only basic determining equation is equation (11).
The no-slip condition means that in the pipe wall velocity fluctuations u' and v' are null, which
implies the nullification of the Reynolds’ stress. Thus, in accordance with expression (9), the shear
wall stress, τw, is given by:
1
τw = −
R
R
∂
 ∂u 
∫ ∂r (τ r ) dr = − ρν  ∂r 
0
(12)
r =R
which depends on the instantaneous distribution of the axial velocities.
The 1D models have the drawback of not considering the transverse structure of the flow,
particularly the law of distribution of velocities by which its entire effect is lumped in the
expression of shear stress in the wall of the pipe. Generally speaking, τw is thus a new unknown
which is added to the problem. Configured in this way, the resolution of the problem requires
knowing an extra relation for the characterization of τw.
3
EVALUATION OF THE UNSTEADY WALL SHEAR STRESS. HYBRID MODEL
In a fully developed steady uniform regime, the equilibrium of forces shows that the average
wall shear stress, τws, is related to the friction slope, Js, as follows:
τ w = τ ws =
ρ gD
4
Js
(13)
In an unsteady flow, particularly a transient one, if the acceleration magnitude is small, it may
be reasonable to accept the validity of the quasi-steady hypothesis. When the accelerations are
too great for the quasi-steady hypothesis to be used (Safwat and Polder, 1973; Latelier and
Leutheusser, 1976; Ramaprian and Tu, 1980), the instantaneous value of the wall shear stress,
τw, requires a more precise evaluation, bearing in mind the effects of the variation in time of
the velocity field. This will require knowing the radial velocity profiles and the Reynolds
stress values. This can be achieved by using a model of greater dimension coupled to a
turbulence model.
A number of researchers have proposed 2D (or quasi-2D) models, using different approaches
and techniques to discretize the respective equations, among whom are Bratland (1986), Ohmi
et al. (1985), Vardy and Hwang (1991), Eichinger and Lein (1992), Silva-Araya and Chaudry
(1997), Pezzinga (1999). The approach adopted in this work, already described in Abreu and
Almeida (2000), makes use of 1D model equations obtained by integrating the axisymmetric
model equations (7)-(8) over the cross-section, using the concept of finite volumes presented by
Patankar (1980). The hybrid methodology proposed (Fig. 1) combines the two models whose
equations exhibit different mathematic behaviour in a connected manner. On the one hand, the
traditional 1D set of hyperbolic quasi-linear equations (which calculates the values of the
piezometric head H and the average velocity V in each section of the pipe on the basis of the
boundary conditions of the problem and the wall shear stress value evaluated by the axisymmetric
model). On the other, the transport equations, parabolic in time and elliptical in relation to space,
of which the momentum equation (8) is an example (the term corresponding to the spatial
gradient of pressure is regarded as a source-term, evaluated by the 1D model). The transport
equations used to model turbulence have exactly the same structure. In the case of turbulent flow,
in fact, a turbulence model has to be added for the calculation of the Reynolds' shear stresses.
Both zero-equation models (like Prandtl's mixed length model) and two-equation models (the k-ε
low Reynolds numbers model) have been used in the model. A more detailed explanation of
this approach and its validation can be found in Abreu (2003).
Input data
Generation of the radial grid
Turbulence Model
Initial velocity profile
Initial steady-state conditions
(t=0)
Velocity distribution
Spatial pressure gradient
Eq. (8)
G(x,t)
u(x,r,t)
Shear stress distribution
Eq. (9)
τ (x,r,t)
1D Model
Eqs. (10)-(11)
H(x,t) e V(x,t)
Wall shear stress
Eq. (12)
τ w (x,t)
Flow perturbation
(induced transient flow)
Νο
Axi-symmetrical model
t = t +∆t
t = tlast ?
Figure 1- The Hybrid model operational concept and structure.
Yes
Stop
4 UNSTEADY WALL SHEAR STRESS MODELLING TECHNIQUES
For one-dimensional modelling, the expression for τw may generally be written as follows:
τ w = τ ws + τ wu
(14)
in which, τws represents the quasi-steady contribution and τwu is the additional contribution due
to unsteadiness. The use of a 1D model requires that appropriate relations are obtained to allow
τwu to be related to other characteristic parameters of the average flow.
A first approach (Vardy et al., 1993; Vardy and Brown, 1995, 1996), consists of defining
approximated weighting functions W, to extend the use of Zielke’s original expression (16).
Vardy and Brown (1995, 1996) proposed for Wa (approximated value of W):
Wa =
ν
1
2 πψ
e
−
ψ
C*
(15)
7.41
 14.3 
, with κ = log10  0.05  , and C* having a minimum value,
κ
Re
R
 Re 
corresponding to the laminar flow, equal to 0.00476. A note of caution is necessary here. Vardy
and Brown (2003b) have confirmed that it was misleading to apply their model to laminar flow
because the assumed velocity profile has a uniform core. It is shown in Section 5.2 that this
causes their model to under-estimate the laminar unsteady skin friction coefficient by about
12%. Vardy and Brown (2003a) have subsequently obtained different values with a better
model. Ghidaoui and Mansour (2002) have recently described an approximate technique for
efficient implementation of Vardy-Brown model via the characteristics’ method.
in which ψ =
2
t and C * =
A second approach, of greater conceptual simplicity, in use since the pioneering work by Daily et
al. (1956) and Carstens and Roller (1959), groups the models in which τwu is agreed to be
proportional to the local instantaneous acceleration of flow in each section of pipe, and this
proportionality is expressed, in the simplest case, by:
τ wu = k
ρ R dV
2
dt
(16)
in which k is an empirical coefficient which is generally regarded as being independent of x
and t. Carstens and Roller (1959), for instance, proposed that k=0.224, based on an
approximate theoretical model. The most successful formulation of this approach is the
expression proposed by Brunone et al. (1991),
τ wu = k3
ρ R  ∂V
∂V 
−a


∂x 
2  ∂t
(17)
later generalized by Pezzinga (2000) and Vítkovský et al. (2000), to take into account the
location in hydraulic system of the exciting source. Later improvements were made by
Loureiro and Ramos (2003).
For a correct choice of k3, usually achieved by ‘trial and error’, the results obtained with (16)
show good agreement with the corresponding experimental results (Pezzinga and Scandura,
1995; Brunone et al., 2000; Bergant et al., 2001), even in cases of liquid column separation
(Bergant and Simpson, 1994). Easy implementation and the good results obtained with this
model explain the widespread acceptance. However, in certain aspects it is not yet been
sufficiently clarified. The first aspect is the prediction of the most appropriate value for the
coefficient k3. The second is the determination of the application limits and the order of
magnitude of errors made. Last, but not the least important aspect, is that of the theoretical
foundation that supports this kind of approximation.
Axworthy et al. (2000) derived a phenomenological equation for the dissipation of energy in an
unsteady flow identical to (17) from thermodynamic considerations. According to the authors,
this would only be valid for transients in which the waterhammer time scale is significantly
shorter than the diffusion time scale. For a transient laminar flow this restriction is equivalent
to
L
R2
<<
a
ν
.
Considering a uniform flow acceleration, for which a closed-form integration of (6) is possible,
Vardy and Brown (1996) have tried to demarcate the validity domain of Brunone’s
approximation. The unsteady component of shear stress, in case of uniform acceleration, tends
asymptotically towards a boundary value, τwuL. The time needed for this to occur is termed
‘rise time’ by the authors, and may be approximated by a dimensionless time:
ψL =
ν
t
2 L
= 3.323 C * . In accordance with Vardy and Brown, therefore, Brunone’s coefficient
R
k3 can be interpreted as an unsteady friction coefficient limit (fuL), given by:
k3 = f uL = 2 C *
(18)
and (17) may provide good approximations for accelerations that occur at temporal intervals
greater than ΨL, but should not be used for accelerations of significantly shorter duration.
Bearing in mind the deduction of the generalized shear stress expressions adapted to 1D
modelling (Almeida and Koelle, 1992), the theoretical approach proposed by Almeida (1981,
1983), validated numerically via the hybrid model by Abreu and Almeida (2000) has been
adopted in this work. In accordance with this approach, the compatibility between momentum
and energy equations, to ensure the uniqueness of the solutions, makes it possible to find the
following relation between τw and J (head slope):
quasi − steady
contribution (τ ws )
τw =
ρ gD
4
J
unsteady contribution (τ wu )
+
ρgD
4
Ju +
dissipative contribution (τ wD )
ρgD 
4
V ∂β 
1 ∂V
( β − 1) g ∂t + 2g ∂t 


(19)
inertial contribution (τ wI )
where β is the Boussinesq or momentum coefficient, resulting from the non-uniform distribution
of instantaneous velocities.
Equation (19), which is a particular case of (14), exhibits the influence of non-uniform
distribution of velocities on each section (β≠1) in the relation τw-J and expresses the fact of τw, in
an unsteady flow, as having dissipation and inertial components. In fact, the last two terms on
the right-hand side of (19) characterize the inertial effects due to the non-uniformity of the
velocity profile at any time and in any section of the pipe (β≠1) and to the redistribution of energy
caused by the temporal change in the velocity profile (β=β (t)). The importance of this correction
is greater the lower the Reynolds’ number. Thus the value of β in a steady regime varies between
4/3≈1.33 in laminar flow, and the limit value of one for flows with very high Reynolds' numbers.
The use of (19) requires knowing the value of β at any time and in any section of the pipe. The
hybrid model gives the temporal evolution of β throughout the transient and therefore allows the
various terms in (19) to be evaluated.
5 DISCUSSION OF RESULTS FOR LAMINAR FLOWS
5.1
Rigid model
It is assumed, as a first approximation, the application of (19) to a rigid transient regime. The
simple hydraulic system shown in Fig. 2(a) is considered. At t=0 changes in valve settings
produce a linear increment in flow velocity in time Tc, as may be seen in Fig. 2(b). The
corresponding change in piezometric head is shown in Fig. 2(c), in accordance with the hybrid
(2D) and quasi-steady (1D) models. The best-known feature is that the quasi-steady 1D model
underpredicts the minimum pressures.
V
V=Vf
∆V
P.L. (t=0)
(b) Prescribed change in flow
V=V0
H0
t=0
8
P.L. (t= )
H
t=Tc
t
8
L dV
g dt
V(t)
(c) Pressure variation at the valve
Quasi-steady model
(a) Schematic flow system
Hybrid model
∆H
k eq. L dV
g dt
:
Figure 2- Imposition of a law of linear variation of flow velocity. Responses of the quasi-steady
(1D) and hybrid (2D) models in the case of a rigid transient laminar regime.
D2 ν
is a significant characteristic parameter
Tc
for this problem. For sufficiently high values of λ, a significant variation occurs in β during the
corresponding variation in flow velocity, according to the results of the numerical simulations
carried out. This can be observed, for example, in Fig. 3(a), for λ=800 (Example 1). The
corresponding variation in the piezometric head is shown in Fig. 3(b). These results correspond
to a linear increase in velocity from 0.05 to 0.15 m/s in time Tc=0.5 s. For a straight 1000 m long
pipe with diameter D=0.02 m and a kinematic viscosity of ν=10-6 m2/s, such variation in velocity
corresponds to a change in the Reynolds’ number from 1000 to 3000.
It has been found that the dimensionless number λ=
Figure 4 shows the mean velocity profiles in the pipe, obtained with the hybrid model,
corresponding to initial (t=0), midpoint (t=0.25s) and final (t=0.5s) times of the prescribed
imposed variation of velocity. Velocity profiles corresponding to the quasi-steady case are also
given. During the period of constant acceleration, the instantaneous velocity profiles were found
to be significantly different from the form corresponding to the quasi-steady case, showing a more
uniform distribution and steeper gradient with the wall. In fact, the unsteadiness is expressed as a
smoothing of the axial velocities throughout the cross-section, and, consequently, in a perceptible
reduction in the value of β during the simulation (from an initial value of 4/3≈1.33 to one close
to 1.10 at its end, as shown in Fig. 3(a)). For these situations, the hypothesis of Carstens and
Roller (1959), mentioned above, in which the velocity profile is assumed to have the parabolic
quasi-steady form throughout the velocity change, is not confirmed.
(a)
(b)
Figure 3- Example 1. Increase in flow velocity from V=0.05 m/s to V=0.15 m/s in Tc =0.5 s,
according to the rigid model and with λ.
(a) Assumed velocity variation and associated computed temporal variation of β (2D model);
(b) Comparison between hybrid model (2D) and quasi-steady model (1D) head oscilations.
Figure 4 - Example 1. Velocity profiles obtained with the hybrid model and the corresponding
profiles assuming the quasi-steady approximation (rigid model with λ).
Linked with this temporal variation of β and with the non-uniformity of the velocity profile, there
is, according to (19), an unsteady inertial component (τwI) of wall shear stress. The histories of
this component and its relative importance in (19) can be seen in Fig. 5. The two parts of the
inertial component, τwI, have been respectively designated τw1 and τw2,
τ wI = τ w1 + τ w 2
with : τ w1 =
ρ gD
4
(β − 1 )
1 ∂V
g ∂t
and
τ w2 =
ρ gD V ∂β
4
2 g ∂t
(20)
As a result of the progressive reduction of β during the time of manoeuvre Tc, described earlier,
the component τw1 of shear stress (Fig. 5(b)) is a decreasing monotonic function with a null
value to the right of the time corresponding to the end of the manoeuvre (t=Tc). The
component τw2, meanwhile, has a behavior almost symmetrical with that described for τw1,
although with slightly lower absolute values. However, contrary to what happens with τw1, it is
not annulled at the end of the manoeuvre, reflecting the fact that the value of β tends
asymptotically towards the value corresponding to the final steady regime. As may be seen in
Fig. 5(b), the function resulting from these two effects, τwI = τw1 + τw2, is, during the course of
the manoeuvre, a growing monotonic function (with positive values), reaching its maximum at
its end. Subsequently, following the evolution of τw2 (τw1=0), tends asymptotically towards
zero. For smaller values of the parameter λ, the value of β can reach its minimum during the
manoeuvre. But it was possible to conclude from the various simulations carried out that the
qualitative trend, mentioned above, is a general tendency in the temporal variation of the
inertial component, τw1.
(a)
(b)
Figure 5- Example 1. Variation of the various components of the pipe wall shear stresses,
obtained with the hybrid model (2D) – rigid model, with λ=800.
(a) Quasi-steady (τωs), global unsteady (τω=τωD+τωI) and dissipation (τωD) components.
(b) Inertial component (τωI) and respective sub-components (τω1 and τω2).
Fig. 5(a) further makes it possible to assess the relative importance of the dissipation and
inertial components of wall shear stress in the unsteady flow. It can be seen that the unsteady
effects predominate in this range of values for parameter λ, with the inertial effects being more
significant than the dissipation effects. Note that this distribution of the relative importance of
the unsteady dissipation and inertial components is a general trend, regardless of the value of λ.
The existence of the limit value, τwuL, in the laminar regime referred to by Vardy and Brown
(1995, 1996) is confirmed. The value in quantitative terms, mentioned by these authors as
corresponding to C*=0.00476, it is not confirmed, for reasons explained near the beginning of
Section 4. According to the considerations described earlier, it can easily be concluded, based on
(16) and (19), that the limit value of k to be used in (16) should be equal to (β - 1)=1/3≈0.33, as
borne out by the numerical results. Actually, the influence of the energy dissipation in the
unsteady flow, for the examples being studied (rigid model), can more or less be taken into
consideration in the 1D model, accepting a value for k, which we shall now call k equivalent
(keq.), taken as the value of parameter k, which, substituted in (16), leads to a maximum
piezometric head variation which coincides with that obtained by the hybrid model (see Fig. 2),
or,
k eq. =
H (t = Tc ) q.s. − H (t = Tc ) hyb.
L dV
g dt
(21)
Figure 6 gives a systematic synthesis of the results of numerical simulations carried out, based
on the hybrid model, corresponding to a uniform acceleration of the flow. The substitution of
the limit value k=0.33 in (16) corresponds to the approximate solution proposed by Shuy
(1995). It should be mentioned, too, the results given in Fig.6 are equally valid in the case
where a linear reduction of flow velocity is imposed.
Accepting the validity of the rigid model, and using the 1D model, a maximum variation
in pressure equivalent to that obtained with the hybrid model can be obtained, if we consider the
following approximate expression:
τ w = τ ws +
ρD
4
keq.
dV
dt
(22)
which corresponds to adding to the wall shear stress in steady flow a supplementary equivalent
shear stress of inertial type.
Figure 6 - Values of keq. as a function of λ.
5.2
Elastic model
It is intended to generalize the methodology employed for the rigid model to the case where the
elastic model is used. Accepting the expression (21), the basic equation (11) can be written as
follows:
∂V
∂H
∂V
4
+g
+
=0
τ ws + keq.
∂t
∂x
∂t
ρD
(23)
with the equation of continuity remaining unchanged.
The presence of the inertial term in (23) changes the equations of compatibility in the method of
characteristics (Axworthy et al., 2000):
dH
a
dV
dx
4
±
1 + keq.
+
=0
τ ws
dt
g
dt
dt
ρgD
with:
dx
= ±
dt
(24))
a
.
1 + keq.
Employing a methodology equivalent to that of Brunone (1991), would give the following
alternative scheme for the equations of compatibility:
dH ±
ρD
4 
a
dV +
τ ws +

g
4
ρ gD 
( 1+ k
eq .

) dV
dx = 0
dt 
−1

(25)
with: dx = ± a dt .
It has been confirmed numerically that the two schemes are equivalent (i.e. both schemes give
identical predicted values) but that the results obtained are not acceptable. However, if the
substantial derivative of the velocity is calculated following the propagation of an elastic wave
(according to the direction of the reflected wave), (25) is equivalent to that proposed by
Brunone, with the wall shear stress expressed as follows:
τ w = τ ws +
ρR
2
( 1+ k
eq .
)
∂V 
 ∂V
−1 
−a

∂x 
 ∂t
(26)
which is identical to expression (17) with: k 3 = 1 + keq. − 1 .
The scheme in this form (or in the alternative form (24) with the derivative of the velocity
calculated following only the reflected wave) gives excellent practical results, especially with
respect to the damping of the pressure fluctuations, as shown in some examples, using the
values of keq. given in Fig. 7.
According to (26) the maximum value of k3 will be: k 3 = 1 + 1 / 3 − 1 ≈ 0.155 and not 0.138, the
limit value proposed by Vardy and Brown (1996). As indicated above, their assumed velocity
profile is more plausible for low Reynolds number turbulent flow than for laminar flow.
The example below (Example 2), already adopted in the analysis using the rigid model (Example
1
1), corresponds to a uniform acceleration in Tc=0.5 s - acceleration pulse of duration ≈ ψ L (Fig.
3
7). Analysis of the results shows that the Brunone provides good results in a range manifestly
outside the zone that Vardy and Brown (1996) held to be the model’s sphere of validity. The
results seem to indicate that when the substantial derivative is considered following the
direction of the reflected wave, an effect similar to that given by the rigid model is obtained. In
fact, an “equivalent” rectangular fluctuation is obtained for the unsteady component, τwu, of the
wall shear stress (Fig. 7(b)).
(a)
(b)
2L
D2 ν
Figure 7- Example 2. Uniform acceleration of flow. Elastic model with
= 800 and Tc =
.
a
Tc
(a) Midpoint piezometric head trace;
histories.
(b) Unsteady component of the wall shear stress
6 DISCUSSION OF RESULTS FOR TURBULENT FLOWS
It could be valid to acknowledge that the quasi-steady hypothesis is more accurate the slower
the variations in velocity, and the higher the Reynolds’ number for flow. As, in most practical
cases, the flow is turbulent, it might be thought that, although from the theoretical and
experimental point of view the problem becomes difficult, from the practical point of view, the
problem would be less relevant, given the less departure in relation to the quasi-steady
hypothesis. In the case where the by now classic data of Holmboe and Rouleau (1967) are
used, the preceding conclusion appears to be confirmed, as shown in the comparisons given in
Fig. 8 with experimental and numerical pressure time histories in the valve section. It may be
observed that there is a certain distance between the results found with the quasi-steady
hypothesis (1D model) and the experimental results (see the main test parameters, Table 1),
although perceptible below what was found in some preceding laminar flow situations. As
with laminar flow, here, too, the results obtained with the hybrid model appear to accurately
predict those obtained experimentally.
Table 1. Main parameters corresponding to the experimental tests described by various authors
Experiment
1- Holmboe and Rouleau
2- Brunone et al.
3- DEC-IST
D(m)
0.0245
0.0983
0.042
L(m)
36
352
108
V(m/s)
0.244
0.65
0.35
Re
7200
63224
14407
a(m/s)
1324
340
291
Tc (s)
0.0
0.06
0.43
P
79.5
2.1
3.9
Figure 8- Comparison of the numerical results and the experimental results of Holmboe and
Rouleau (1967) following sudden valve closure.
However, when the same comparison is made (Fig. 9) with the results presented by Brunone et
al. (2000), corresponding to the rapid closure of a spherical valve (see the main parameters for
experimental test 2, Table 1) it is concluded that the dissipation observed experimentally is
considerably greater than that suggested by the hybrid model. But where the model proposed
by Brunone in which τw is estimated by (17) or (26), a value for k3 of the order of 0.1 has to be
used to obtain the same dissipation. This value is clearly greater than what would be expected
according to the analysis carried out by different authors (Vardy and Brown,1996; Pezzinga,
2000), since k3 will decreased rapidly as Re increases.
For this test, Brunone et al. (2000) also give average instantaneous velocity profiles in the same
section, immediately upstream of the valve, for different times. Some of the profiles presented
are of a strongly asymmetrical nature, as Brunone et al. (2000) mention. This experimental
evidence, plus the fact that, in the variable regime, the values for the Reynolds’ stresses are not
necessarily equal to those corresponding to a steady uniform flow in the same pipe and with an
instantaneous Reynolds' number, may well explain the different behavior exhibited by the
experimental and numerical results.
Figure 9- Comparison of the numerical results achieved with the 1D model (quasi-steady
hypothesis) and with the hybrid model, with the experimental results reported by Brunone et al.
(2000) corresponding to a section of pipe next to the valve.
Recently, Ghidaoui et al (2002) have published a study whose main findings are: (1) the
solution in transient regime is not very sensitive to the magnitude and distribution of the
turbulent viscosity in the central region of the pipe; the flow zone adjacent to the wall exerts a
dominant influence; (2) the relation between the time scale for turbulent diffusion
( 2 D / 2uτ = 2 D / f V ) and the corresponding scale of elastic pressure wave propagation
(L/a) define an dimensionless parameter, P, suitable for defining the limit of validity of the
quasi-steady assumption in relation to modeling the turbulence; for values of P>>1 the quasisteady assumption may be regarded as valid as long as, for values of P of order unity the
structure and intensity of the turbulence undergo profound change during the transient
phenomenon, and the aforementioned hypothesis is highly questionable; (3) in cases where
flow asymmetry has been observed experimentally, the difference between the numerical and
experimental results grows exponentially. The conclusions reported by Ghidaoui et al. (2002)
seem to explain the results obtained earlier. In fact, calculating the values of parameter P
(Table 1) corresponding to the experimental tests conducted by Holmboe and Rouleau (1967)
and Brunone et al. (2000) we get 79.5 and 2.1, respectively. Preliminary experimental results
obtained recently in a research project (DEC-IST, 2002) also seem to point in the same direction,
as can be seen from the comparison made in Fig. 10. For the conditions of this test (Table 1), a
value of P=3.9 was obtained which, according to the above criterion, makes it possible to
conclude that this is a case where the conditions for applying the quasi-steady hypothesis vis-à-vis
turbulence modelling are highly questionable. It should also be mentioned that the preliminary
results for velocity profiles corresponding to this experimental test suggest some asymmetry, as in
the experimental work by Brunone et al. (2000).
Figure 10- Comparison of the numerical and experimental (DEC-IST) results.
7 CONCLUDING REMARKS
By integrating the governing axisymmetric flow equations over the cross-section of the pipe, the
full effect of the transverse structure of the flow, that is, the effects arising from the nonuniformity of the velocity profile, is concentrated in the expression of the pipe wall shear
stress, τw. As a consequence, in transient flows the unsteady contribution of τw constitutes a
new unknown that is added to the problem.
It has been shown theoretically, and confirmed by means of the hybrid model, that this
unsteady contribution has two components: a dissipation component and an inertial
component, and it is this latter that is, as a rule, mostly predominant. The fundamental
contribution of this inertial component is proportional to the instantaneous local acceleration of
flow, expressed in the simplest case by (16). This fact explains why, in some situations, the
exclusive consideration of this component might constitute a good approach to the problem.
This has been shown for the case of a linear variation of velocity in laminar flow, when the
time of velocity variation is not long in relation to the characteristic time of the process of
radial diffusion of the quantity of movement from the pipe wall, that is, when the
dimensionless parameter λ takes low values, close to or lower than the unity. This fact alone
validates the traditional approach, based on (16), for a given range of values of λ, and it also
seems to bolster Vardy and Brown's analysis, which refers to the validity of that approach.
But for laminar flow and using the rigid model, it has been shown that the joint effect of the
unsteady contribution of τw (τwu) could be regarded as an approximated form via (16), as long
as a value for k, known as keq., variable with λ is considered. More significant, however, is the
fact that these same values of keq, when used in the corresponding expression for the elastic
model (26), allow an excellent modeling of the damping of pressure fluctuations, and, in these
circumstances, validate Brunone’s model for the whole range of values for λ.
In a turbulent flow, although the same kind of approach can be used (with keq dependent on the
Reynolds’number), the question of the validity of the quasi-steady hypothesis immediately
arises in order to turbulence modelling. In conclusion, for more meaningful progress in the
numerical modelling of turbulent transient flows, there will first have to be full experimental
confirmation of the preliminary conclusions, mentioned above, with particular reference to the
domain over which the quasi-steady and flow symmetry approximations remains useful. This
goal can only be achieved by carrying out a larger series of experimental tests, covering a
wider range of conditions, based on an integrated approach involving both the numerical and
experimental aspects.
8 ACKNOWLEDGEMENTS
This work was developed as part of a research project funded by the Portuguese National
Science Foundation (PRAXIS XXI, 3/3.1/CEG/2503/95).
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NOTATION
The following symbols are used in this paper:
A
a
c
D
f
fuL
gx, gr
=
=
=
=
=
=
=
cross-sectional area of pipe;
pressure wave celerity in a pipe;
isentropic speed of sound in fluid;
pipe inner diameter;
pipe Darcy-Weisbach friction factor;
limiting value of the unsteady friction factor;
acceleration of gravity according to x and r;
H
J
k
k3
keq
L
M
P
p
R
r
Re
Tc
t
V
u
v
u ′ v′
x
z
W
Wa
Wr
= piezometric head;
= head loss by unit pipe length;
= empirical coefficient of unsteady friction 1D models;
= coefficient of unsteady friction model of Brunone (eq. 33);
= coefficient defined following equation (21);
= pipe length;
= Mach number;
= dimensionless parameter;
= fluid pressure;
= pipe inner radius;
= radial coordinate;
= Reynolds number;
= time of manoeuvre;
= time;
= mean axial velocity;
= axial velocity component;
= radial velocity component;
= Reynolds stresses
= axial co-ordinate;
= elevation of the pipe axis;
= weighting function;
= approximate weighting function;
= pipe wall radial displacement;
β
= Boussinesq or momentum coefficient;
ν
= kinematic viscosity;
ρ
= fluid density;
τ
= shear stress;
τw
= wall shear stress;
τws = quasi-steady wall shear stress;
τwu = unsteady wall shear stress;
τwD = unsteady dissipative component of wall shear stress;
τwI = unsteady inertial component of wall shear stress;
τw1,τw2= two parts of the unsteady inertial component of wall shear stress;
θ
= angle the pipe’s axis makes with the horizontal; cylinder polar coordinate;
D2 ν
λ
= dimensionless number,
;
Tc
ΨL = non-dimensional time required to reach the limiting shear stress (rise time)