![Illustration of the experimental test rig [6]. The flow is driven by gravity from the upstream tank 2 O to the downstream tank 6 O, while being counteracted by the losses in the components of the system. The downstream valve 5 O is closed and opened](https://www.researchgate.net/profile/Jonathan-Fahlbeck/publication/357991477/figure/fig2/AS:1114583246876672@1642748762454/Illustration-of-the-experimental-test-rig-6-The-flow-is-driven-by-gravity-from-the_Q320.jpg)
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Volume 2 [Full Papers], Pages 1–12
ISSN: 2753-8168
A HEAD LOSS PRESSURE BOUNDARY CONDITION
FOR HYDRAULIC SYSTEMS
JONATHAN FAHLBECK∗, H˚
AKAN NILSSON AND SAEED SALEHI
Division of Fluid Dynamics, Department of Mechanics and Maritime Sciences, Chalmers University of
Technology, Gothenburg SE-412 96, Sweden
Email address:fahlbeck@chalmers.se
DOI:10.51560/ofj.v2.69
Version(s):OpenFOAM v1912, v2006, v2012, v2106, v2112
Repo:–
Abstract. Despite the increase in computational power of HPC clusters, it is in most cases not possible
to include the entire hydraulic system when doing detailed numerical studies of the flow in one of the
components in the system. The numerical models are still most often constrained to a small part of the
system and the boundary conditions may in many cases be difficult to specify. The headLossPressure
boundary condition is developed in the present work for the OpenFOAM open-source CFD code to
include the main effects caused by a large hydraulic system onto a component in the system. The
main motivation is to provide a boundary condition for incompressible hydraulic systems where known
properties are specified by the user and unknown properties are calculated. This paper is a guide to
the developed headLossPressure boundary condition. It is based on the extended Bernoulli equation
to calculate the kinematic pressure on the patch. An arbitrary number of minor and friction losses are
considered to describe the system in terms of head losses. The boundary condition also provides the
opportunity to specify the head (difference in height) in relation to a reference elevation. System changes
during operations are modelled through Function1 variables, which enables time-varying inputs. The
developments are validated against experimental test data, where the varying head between two free
surfaces and a valve closing and opening sequence are modelled with the boundary condition. The main
effects of the system are well captured by the headLossPressure boundary condition. It is thus a useful
and trustworthy boundary condition for incompressible flow simulations of components in a hydraulic
system.
1. Introduction
Computational Fluid Dynamics (CFD) is inherently computationally demanding and requires large
computer resources [1]. The demands of the numerical simulations increase drastically with the size of
the computational domain. Therefore, often the computational domains are as much as possible limited
to the region of the studied component. In hydraulic system simulations, the dynamics of the system that
the studied component is connected to is often neglected, and approximate (unknown) static boundary
conditions are in most cases applied for convenience. This puts great limitations on the usefulness and
accuracy of the results. Simplified 1D methods, such as the Method of Characteristics (MOC) [2] or
Method of Implicit (MOI) [3], are sometimes coupled to full 2D/3D CFD simulations to include the
effects of the system. This requires access to a MOC/MOI solver and a coupling interface between
the CFD solver and the MOC/MOI solver. With MOC/MOI, the pressure waves in the system are
included, which in the case of hydraulic systems requires the pressure waves to be resolved also by the
CFD solver. This puts enormous constraints on the time step due to the high sound speed. The aim
of the present work is to provide a boundary condition for incompressible 1D hydraulic systems that is
implemented using OpenFOAM libraries and can be linked to OpenFOAM using the standard linking
procedures of OpenFOAM. Built on an incompressible assumption, it allows larger time steps compared
to an compressible formulation, since no pressure waves need to be resolved. The present work thus
provides a cost-efficient method to model the main effects from 1D incompressible hydraulic systems
with one single pressure boundary condition in OpenFOAM. The motivation behind the work is the
need to simulate a component in detail where both the pressure difference between the inlet and outlet
of the computational domain and the flow rate are highly dependent on the system. Hence, the flow
∗Corresponding author
Received: 26 October 2021, Accepted: 10 January 2022, Published: 20 January 2022
2022 The Authors. This work is an open access article under the CC BY-SA 4.0 license
1
2 JONATHAN FAHLBECK∗, H˚
AKAN NILSSON AND SAEED SALEHI
rate and pressure difference are both calculated with the developed boundary condition as a part of the
CFD solution. This is accomplished by a boundary condition that considers the characteristics of an
arbitrary number of components of the system which the studied component is connected to. The use
of time-varying variables, i.e. Function1, allows the boundary condition to be utilised for transients,
where some of the system characteristics change over time. The boundary condition is implemented as a
subclass to the fixedValue type. It is developed based on the totalPressure boundary condition and is
a further development of the initial work made by Fahlbeck [4]. The implementation is validated against
measured data for an experimentally studied test case. In addition to the description and validation of
the new boundary condition, the paper also provides a general description on how to incorporate head
losses at the patches of the computational domain in OpenFOAM. The provided library can be employed
by anyone who would like to model a specific component of a large hydraulic system while considering
the effect of other components. The implementation was originally developed for OpenFOAM v2012.
It has however been tested for several recent OpenFOAM versions (v1912–v2112) and for a number of
incompressible solvers, including simpleFoam,pisoFoam,pimpleFoam and interFoam (only one phase at
the patch is allowed).
2. Theoretical background
The essence of the developed headLossPressure pressure boundary condition is explained in this
section through mathematical expression and theoretical assumptions. We start by considering Bernoulli’s
energy principle [5]. It states that the energy of the flow is constant along a streamline (if no losses are
considered) as
p∗+u2
2+gz = constant.(1)
Here p∗is the kinematic pressure (p∗=p/ρ, where ρis the fluid density), uis the velocity, gis the
gravitational acceleration, and zis the position in the direction of the gravitational acceleration with
respect to some reference level. The energy equation can be further extended to include the head losses
along a streamline between an upstream (subscript u) and downstream (subscript d) point as
p∗
u+u2
u
2+gzu=p∗
d+u2
d
2+gzd+ ∆p∗
m+ ∆p∗
f.(2)
Here ∆p∗
mand ∆p∗
frepresent head losses due to local occurrences in the flow path (minor) and friction
from the wall (major), respectively.
Consider the simple hydraulic system shown in Figure 1. The simulated computational domain is
located between points 2 and 3, and the headLossPressure boundary condition updates and adjusts the
pressure at patches 2 and 3 by applying the extended Bernoulli equation between points 1–2 and 4–3,
respectively, as
p∗
2=p∗
1+1
2u2
1−u2
2+g H12
±
z1−z2
−(∆p∗
m+ ∆p∗
f),(3)
p∗
3=p∗
4+1
2u2
4−u2
3+g H34
±
z4−z3
+ (∆p∗
m+ ∆p∗
f).(4)
If the reservoirs at 1 and 4 are large, it is assumed that the flow velocity is zero at the free surface, which
is a common assumption. The extended energy equation, Eq. (2), for the patches at 2 and 3 can thus be
expressed as
p∗
2=p∗
1−u2
2
2+gH12 −(∆p∗
m+ ∆p∗
f),(5)
p∗
3=p∗
4−u2
3
2+gH34 + (∆p∗
m+ ∆p∗
f).(6)
The head losses are subtracted at p∗
2since it is downstream of p∗
1, and they are added at p∗
3since p∗
4is
upstream of p∗
3, which is in accordance with Eq. (2). It should be noted that the sign of the loss terms
can be made dependent on the flow direction at the boundaries, making the implementation independent
of the flow direction. There may for instance be a pump in the CFD region, pumping the water from the
lower reservoir to the upper reservoir. This is taken care of in the present implementation.
Local head loss occurrences in the flow path are commonly referred to as minor head losses, and friction
losses from the wall as major head losses. In this work the terms minor and friction are used for those
HEAD LOSS PRESSURE BOUNDARY CONDITION 3
Lower
Upper
CFD
1
23
4
12
34
3D CFD
Boundary condition Boundary condition
Figure 1. Example of a hydraulic system with upper and lower reservoirs, numerical
(CFD) domain, and pipelines. The flow is here assumed to be from the upper to the
lower reservoir as indicated by the arrows (could be reversed if there is a pump in the
CFD region), and the markers 1–4 are in accordance with Eqs. (3)–(6).
head losses. Minor head losses are caused by sudden changes in the flow path or separation, for instance
bends, valves, cross-sectional area changes, etc. The minor losses depend on the dynamic pressure and a
minor loss coefficient, k. An arbitrary number of minor head losses can be added as
∆p∗
m=
Nm
X
i=1
ki
u2
i
2.(7)
Here Nmis the number of minor losses. The minor loss coefficient of each component, ki, is usually found
as tabulated values in textbooks, provided by the manufacturer, calculated with numerical models, or
estimated based on a set of experimental data.
The head loss due to friction in an arbitrary number of pipes can be summed up as
∆p∗
f=
Nf
X
i=1
fi
Li
dh,i
u2
i
2,(8)
where Liand dh,iare the length and hydraulic diameter of each pipe, respectively. fiis the friction loss
coefficient of each pipe, and Nfis the number of friction losses. In a laminar case, fis calculated as a
function of the Reynolds number, as
f=64
Re,(9)
where Re = udh/ν, in which νis the kinematic viscosity. However, if the flow is turbulent (Re >2300),
fis solved iteratively using the implicit Colebrook equation [5],
1
f1/2=−2.0 log10 ϵ/dh
3.7+2.51
Ref1/2,(10)
where ϵis the surface roughness. The explicit Haaland equation [5], given by
f1/2≈1
−1.8 log10 ϵ/dh
3.71.11
+6.9
Re ,(11)
can be used as an initial guess before solving the more complex Colebrook equation (Eq. (10)).
Hence, a minor head loss is only a function of the local flow velocity and the minor loss coefficient,
whereas a friction head loss is a function of the local flow velocity, the pipe length and diameter, and
the surface roughness. If all the losses are arranged sequentially (without bifurcated flow paths), as is
presently assumed, the local flow velocity can be calculated from the flow rate and the local cross-sectional
area.
4 JONATHAN FAHLBECK∗, H˚
AKAN NILSSON AND SAEED SALEHI
3. Implementation of the boundary condition
To reach the goal with the headLossPressure boundary condition, one must be able to account for any
number of head losses in the system. In section 4.3 an example of user-inputs is given. To enable the user
to specify an arbitrary number of minor and friction loss factors, the user-inputs minorLossFactors and
frictionLossFactors are declared according to Listing 1. Each of the two variables is constructed as a
List of Tuple2 type, where Tuple2 has the capability of storing two objects. For the minor losses, the
Tuple2 consists of a vector2D and a word. This ensures that the user can supply a list of the necessary
information in the appropriate format (hydraulic diameter, minor loss coefficient), and a name of the
particular loss. A similar construction is made for the friction loss factors, the difference being that the
Tuple2 consists of a vector and a word. This enables the user to supply the list of friction loss factors in
the following format (hydraulic diameter, surface roughness, pipe length), and the name of the friction
loss. At least one minor and friction loss factor must always be specified for the boundary condition to
work. Specifying a minor loss coefficient of zero will disable the calculation of the minor loss. For the
friction loss factor, the pipe length must be set to zero to disable all effects due to friction. The name of
each loss is used by Info and Warning statements during run time, and it helps the user to keep track of
the individual losses.
//− Tuple2 list of the minor loss factors ((d, k) name)
typedef Tuple2<vector2D, word>indexedVector2D;
const List<indexedVector2D>minorLossFactors ;
//− Tuple2 list of the friction loss factors ((d, epsilon, L) name)
typedef Tuple2<vector, word>indexedVector;
const List<indexedVector>frictionLossFactors ;
Listing 1. Declaration of the minor and friction loss factors
There are additional user-inputs that are required (and some that are optional) when applying the
boundary condition, and in Table 1 the full list of available inputs is shown. The variable pFar corresponds
to the kinematic pressure ‘far’ from the patch in the hydraulic system, e.g. up or downstream in the pipe
or at a free surface. The variable HFar is used to calculate the hydrostatic pressure, gH, which typically
is a free surface. Both pFar and HFar contribute with a constant pressure on the patch. If a gravity
based solver (e.g interFoam) is used, the HFar at the patch must be in reference to the head specified
in the hRef dictionary. In the case of a non-gravity based solver (e.g. simpleFoam or pimpleFoam) the
head must be in reference to the system. Hence, if the headLossPressure is used at multiple patches,
the pressure that drives the flow will be correct as long as the same reference level is used. However, the
hydrostatic pressure on a specific patch may be incorrect if the computational domain has an extension
in the direction of the gravity acceleration. The same is true if the headLossPressure is used only on
one patch, the head specified for this single patch must be the head of the system to get the correct flow
rate of the system.
The three variables dP,minorLossFactors, and frictionLossFactors are used to calculate the head
losses. The hydraulic diameter of the patch, dP, and the hydraulic diameter specified for each head loss
is used together with the flow rate at the patch to calculate the velocity that is needed to calculate each
head loss. The entries minorLossFactors and frictionLossFactors are for the minor and friction loss
factors, respectively, where the details are explained in the Table 1 footnotes.
The special variable kDynamic is a single dynamic minor loss coefficient, k, with Function1 capabilities,
since the list of minor loss coefficients in minorLossFactors cannot use Function1. This is useful when,
for instance, modelling a valve closing or opening sequence with the boundary condition. Due to the fact
that the valve may be located in a pipe with a different cross-sectional area than the patch, the variable
dkDynamic must also be supplied to re-scale to an appropriate flow velocity for the loss.
The boundary condition can also model the filling or emptying of a reservoir using the parameters
flowRate and Ar, as
HFar =Ho+Qr−Qp
Ar
∆t. (12)
Here HFar is the head of the reservoir at the new time step, Hois the head of the reservoir at the
previous time step, Qris the user-supplied flowRate to the reservoir, Qpis the flow rate out of the
reservoir (calculated from the volumetric flux at the patch), and Aris the user-supplied surface area of
the reservoir, Ar. The calculation of a new HFar is only made if the user supplies the flowRate and Ar.
HEAD LOSS PRESSURE BOUNDARY CONDITION 5
Table 1. Available user-inputs for the headLossPressure boundary condition.
Variable Description Required Dimension Default Function1
pFar Kinematic pressure far from patch, p∗
Far yes m2/s2- no
HFar Elevation far from patch , HFar yes m - no
dP Hydraulic diameter of patch, dh,p yes m - no
minorLossFactors Minor loss factors: (d∤
h,l,k†† ), name‡‡ yes (m,-), - - no
frictionLossFactors Friction loss factors: (d∤
h,l,ϵ∥,L⋆), name‡‡ yes (m,m,m), - - no
kDynamic Dynamic minor loss coefficient, kno - - yes
dkDynamic†Hydraulic diameter of kDynamic,d∤
h,l no m - no
flowRate Flow rate to the reservoir, Qrno m3/s - yes
Ar‡Reservoir surface area, Arno m2- no
dFar§Hydraulic diameter far from patch, dh,Far no m 0 no
UVelocity field name no - U no
phi Flux field name no - phi no
g¶Gravitational acceleration no kg m/s29.81 no
Tol Tolerance for the Colebrook equation no - 10−6no
Nitr Max iterations for the Colebrook equation no - 20 no
fDiff Max difference between fand fInitial no - 0.05 no
†dkDynamic is required if kDynamic is supplied, ‡Ar is required if flowRate is supplied, ∤hydraulic diameter at the loss,
†† minor loss coefficient, ‡‡ name of the loss, ∥surface roughness, ⋆pipe length, §default is 0 and this gives that the far
velocity is assumed to be zero (e.g. large reservoir), ¶only used if a non-gravity based solver is used (otherwise the g
dictionary is used)
The head cannot be allowed to vary within a time step, thus HFar is only updated at the first inner-loop
if a transient solver (e.g. pimpleFoam or pisoFoam) is used. Qpis in this case from the final corrector
loop at the previous time step.
If the headLossPressure boundary condition is used within a system where there are no free surfaces
(e.g. a pipe system), the velocity far from the patch cannot be assumed to equal zero. In this case the
variable dFar is used to calculate the velocity far from the patch as
uFar =Ap
AFar
uavg ⇒uFar =dh,p
dh,Far 2
uavg.(13)
Here dh,p is the hydraulic diameter of the patch, dP,dh,Far is the hydraulic diameter far from the patch,
dFar, and uavg is the average velocity of the patch (calculated from uavg =|ϕ/A|patch , where ϕis the
volumetric flux of the patch). It is thus assumed that the local velocity far from the patch can be scaled
with the ratio of hydraulic diameter, which is valid if the same type of cross-section is used. With this
manipulation, the flow rate of the system will be given as part of the solution.
3.1. Calculation of head losses. The calculation of the sum of minor head losses is governed by Eq. (7),
and the code that handles the calculation is shown in Listing 2. At line 2 the minor head loss, ∆p∗
m, is
defined and initiated either based on the present value of the kDynamic entry (if available) or equal to
zero. Line 9 shows the general calculation of the ith minor head loss as
∆p∗
m,i=ki
2"uavg dh,p
dh,l,i 2#2
.(14)
Here uavg is the average velocity on the patch. The expression (dh,p /dh,l)2is used to re-scale the velocity
for the specific minor loss, where dh,p and dh,l are the hydraulic diameter of the patch and loss, respectively.
It is thus assumed that the local velocity of the minor loss is only dependent on the relation in hydraulic
diameter between the patch and the loss in square, uloss = (dh,p/dh,l)2uavg . Eq. (14) is used for all the
user-supplied minor loss factors to sum up the combined head loss due to minor effects.
The head loss due to friction in the pipes is calculated in a similar fashion as the minor losses, however
now according to Eq. (8). The main difference here is that the friction loss coefficient, f, needs to be
calculated iteratively using Eq. (10) for a turbulent flow case. The process of how fis estimated is shown
in Figure 2, and the procedure is repeated for all the individual friction losses. The first step is to check if
the flow is laminar or turbulent, where a critical Reynolds number of 2300 is used. If the flow is laminar,
fis immediately found by Eq. (9). In the turbulent case, the Haaland equation (Eq. (11)) is used as the
initial guess when estimating the friction loss coefficient by iterating the Colebrook equation (Eq. (10)).
6 JONATHAN FAHLBECK∗, H˚
AKAN NILSSON AND SAEED SALEHI
1scalar kDynamic = kDynamic ? kDynamic −>value(this−>db().time().timeOutputValue())
: 0;
2scalar dpMinor = kDynamic ? kDynamic*sqr(Uavg*sqr(d P /dkDynamic ))/2 : 0;
3scalar dMinor;
4scalar k;
5forAll(minorLossFactors , i)
6{
7dMinor = minorLossFactors [i].first().component(0);
8k = minorLossFactors [i].first().component(1);
9dpMinor += k *sqr(Uavg *sqr(d P / dMinor))/2;
10 }
Listing 2. Calculation of the minor head losses.
If the relation between the estimated value, f, and the previous value, fo, is above a tolerance Tol, then
the latest fis used as a new guess for the next iteration. The default value of Tol is 10−6. This process
is repeated a maximum of Nitr times to avoid slowing down the simulation, where the default value
of Nitr is 20. If |f−fo|/fo≤Tol never happens, the latest fis passed on to the next step. Usually,
a converged fis obtained within the default tolerance after just a few iterations. To ensure that the
iteration process has not diverged, the final fis compared to the initial fi. The solution obtained by the
Haaland equation should only differ from that by the Colebrook equation by ±2 % [5], so if the difference
is more than fDiff (5 % is the default value) the initial value from the Haaland equation is used and an
Info statement is printed to the command window.
3.2. Patch pressure calculation. Once all the minor and friction head losses are calculated, the kine-
matic pressure on the patch is updated according to Eq. (3) and (4) for inflow and outflow faces, respec-
tively. Listing 3 shows the piece of code that calculates and updates the pressure for the patch faces.
Note that Up and phip are velocity and volumetric flux per face on the patch, respectively, whereas Uavg
and sumPhi_ are integrated velocity and volumetric flux over the patch. The function neg returns 1 if
the variable is negative, and the function pos returns 1 if the variable is positive only, otherwise both
return 0. The mathematical expression of Listing 3 is given by
Re ≤2300 f,
Eq. (9)
fo=fi,
Eq. (11)
f=f(fo),
Eq. (10)
|f−fo|
fo≤Tol
fo=f
|f−fi|
fi≤fDiff
f=fi
Finish
(return f)
Start
no (max Nitr times)
yes yes
no
no yes
Figure 2. Flow chart of iteration procedure to obtain the friction loss coefficient f. Here
Tol is a tolerance, Nitr is the maximum number of iterations for the Colebrook Eq. (10),
and fDiff is the maximum allowed difference between the Haaland Eq. (11) and Cole-
brook Eq. (10), index i and o are for initial and old, respectively.
HEAD LOSS PRESSURE BOUNDARY CONDITION 7
operator==
(
pFar + gHFar + sumPhi /mag( sumPhi )*(dpFriction + dpMinor)
− 0.5*neg(phip)*magSqr(Up) − 0.5*pos(sumPhi )*magSqr(Uavg)
+ 0.5*UFarScale *magSqr(Uavg)
);
Listing 3. Patch pressure calculation
p∗
p,i=p∗
Far
±
I
+gHFar
´¹¹¹¹¹¸¹¹¹¹¶
II
−X1(ϕsum) (∆p∗
f+ ∆p∗
m)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
III
−0.5X2(ϕi)|up,i|2
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹ ¶
IV
−0.5X3(ϕsum)|uavg|2
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹ ¶
V
(15)
+ 0.5u2
Far
´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¶
VI
,
where the functions Xare defined as (note that ϕsum ̸= 0)
X1(ϕsum) = (−1 if ϕsum <0
1 if ϕsum >0, X2(ϕi) = (1 if ϕi<0
0 if ϕi≥0, X3(ϕsum) = (0 if ϕsum <0
1 if ϕsum >0.(16)
Here ϕis the volumetric face flux of the boundary patch, subscript sum represents the summation of all
patch faces and index icorresponds to face ion the patch. Terms I and II denote the kinematic pressure
and hydrostatic pressure far from the patch, respectively. To model the head-rise of a free surface in a
reservoir or tank, or to find the head of the system, HFar is updated once per time step/iteration with
Eq. (12) (if flowRate and Ar are supplied in the pdictionary for the patch). Term III is where the head
losses are added or subtracted according to Eq. (2). They are subtracted for an inflow patch since the
losses are located upstream of the patch. For an outflow patch, the losses are added to the patch since the
losses are downstream of the patch. Terms IV and V give the dynamic pressure contribution. For inflow
faces (negative ϕi) term IV is active, and for an outflow patch (positive ϕsum) term V is switched on.
Thus, the boundary condition is set on a face-by-face basis so that the kinematic pressure on the patch
varies with the velocity for inflow faces with term IV. This is to ensure the preservation of continuity in
combination with a proper velocity boundary condition, and it is the same procedure as is used in the
totalPressure boundary condition. If there are only outflow faces on the patch, term IV is deactivated
on the patch and a uniform dynamic pressure is obtained with term V. This means that for a patch
with mixed flow (both inflow and outflow faces) the pressure will be nonphysically represented with this
boundary condition. To alert the user if reversed flow is encountered on the patch, a warning message is
printed in the command window. The error produced by the boundary condition will increase if adverse
pressure gradients are expected on parts of the patch. However, if the flow direction is well defined, a
mixed flow on the patch may indicate that the flow velocity is small or close to zero. If that is the case,
the error produced by the boundary condition will also be small. This is because the average velocity,
uavg, is the summation of the volumetric face fluxes divided by the patch area, hence a small volumetric
flow equals a small error. Term VI is the velocity far from the patch and it is calculated from Eq. (13).
The variable UFarScale_ in Listing 3 is the relation in hydraulic diameter of the patch, dh,p, and far,
dh,Far, to the power of four, (dh,p/dh,Far)4. As mentioned in Table 1, if the variable dFar is not specified
in the boundary condition dictionary, or provided as zero, the variable UFarScale_ is set to zero. In that
case the patch pressure calculation is given by Eqs. (5) and (6), for an inlet and outlet patch, respectively.
However, if dFar is specified, the patch pressure calculation is according to Eq. (3) for inflow and Eq. (4)
for outflow.
4. Validation Case
The implementation and functionality of the developed headLossPressure boundary condition is
validated with experimental test data reported by Petit et al. [6] and results from numerical simulations
by Wang et al. [7].
8 JONATHAN FAHLBECK∗, H˚
AKAN NILSSON AND SAEED SALEHI
4.1. Experimental rig and computational domain. The experimental test rig is illustrated in Fig-
ure 3. The flow enters the upstream tank 2at a steady discharge of 50 l/s. With a partly closed
upstream valve 3and a fully open downstream valve 5, the free surfaces of the upstream and down-
stream tanks stabilise at roughly 3 m and 0.5 m above the centreline of the pipe 4, respectively. The
flow in the system is thus driven by the head difference between the upstream and downstream tanks,
and counteracted by the losses in the components of the system. A transient sequence was studied ex-
perimentally, where the downstream valve 5closed and opened. The pipe 4had a length of 10 m and
a rectangular cross-section of 0.2 by 0.25 m (width×height). The downstream valve 5and a pressure
sensor were placed at 2 m and 3.245 m upstream the downstream tank 6, respectively.
Figure 3. Illustration of the experimental test rig [6]. The flow is driven by gravity
from the upstream tank 2to the downstream tank 6, while being counteracted by the
losses in the components of the system. The downstream valve 5is closed and opened
to give a transient sequence.
To validate the headLossPressure boundary condition a small 1D computational domain is placed
in the rectangular pipe between markers 4and 5in Figure 3. The computational domain is shown in
Figure 4. Being a 1D computational domain, it has no losses and does therefore not contribute to the
system balance. The entire system is thus represented in the headLossPressure boundary conditions
at each side of the computational domain, which is the subject of the present validation. The boundary
condition is fully capable of being used for a 3D computational domain, and in that case the domain is
part of the system balance. The test rig was originally designed to validate simulations of water hammer
and pressure oscillations in the system. The current development is restricted to incompressible flow,
which limits the results to the main variations in the system and excludes the pressure waves.
Figure 4. 1D computational domain. The flow is from the left to right patch, and all
sides are treated with the empty boundary condition.
4.2. Operational sequence. The opening of the downstream valve ( 5in Figure 3) in the experiment
varied during a transient sequence as shown in Figure 5. The valve was fully open for the first five seconds.
Then, the valve closed linearly between 5–10 s, and it was remained fully closed for eight seconds. The
valve was then opened linearly between 18–23 s.
HEAD LOSS PRESSURE BOUNDARY CONDITION 9
Figure 5. Valve opening v(values at left axis) and loss coefficient kvalve (values at right
axis) as a function of time. Full operational sequence (left plot) and zoomed views of
the smoothed parts of the operational sequence at 10 s (small upper right plot) and at
18 s (small lower right plot). The valve is fully open at 250 mm, and closed at 0 mm.
To model the closing and opening of the valve, the Function1 entry kDynamic of the headLossPressure
boundary condition at the outlet patch of the computational domain is used to prescribe the loss coef-
ficient of the valve at different opening positions. According to the work by Wang et al. [7], the minor
loss coefficient of the valve is estimated as a function of the opening as
kvalve(v) = exp(−2.1469 ln(v) + 12.1624) −1.3614,(17)
where vis in millimetres (250 mm is fully open, and 0 mm is closed). The subtraction of 1.3614 ensures
that the expression is zero as the valve is fully open. In the numerical simulation it is not possible to close
the valve entirely since the kvalve function in Eq. (17) tends to infinity. Thus, a minimum valve opening
of 2 mm is used in the simulation. This gives the largest value of the dynamic minor loss coefficient,
kvalve(2) ≈4.3×104, as shown by the kvalve in Figure 5. The valve opening curve in Figure 5 (left) has a
smooth transition at times 10 s (small upper right plot) and at 18 s (small lower right plot) for numerical
stability, which yields a corresponding smooth transition for the minor loss variation.
The discharge into the upstream tank ( 2in Figure 3) remained constant at 50 l/s during the entire
sequence. Thus, the free surface in the upstream tank was rising during the time that the downstream
valve was not fully open.
4.3. Numerical set-up. The pimpleFoam solver is used for the numerical simulation in the 1D compu-
tational domain shown in Figure 4. The mesh is constructed using the blockMesh utility and contains
20 cells in the streamwise direction. No turbulence model is used in the simulations. The linear scheme
is used for gradients and convection terms for all variables. The backward scheme is used for temporal
discretisation, with a fixed time step of 10−3s. The headLossPressure boundary condition is applied
at both the inlet and outlet patches, where the system components at each side are modelled in the
boundary condition. The results are thus in this case entirely dependent on the developed boundary
condition.
The inputs to the headLossPressure boundary condition at the inlet and outlet patches are shown
in Listings 4 and 5, respectively. The variable pFar is given as uniform 0 at both the inlet and the
outlet. This is because the atmospheric pressure is the same at the free surfaces of the upstream and
downstream tanks, and hence just a reference pressure. The test rig shown in Figure 3 indicates no water
levels. However, the generic system shown in Figure 1 has a similar layout as the experimental test rig.
In Figure 1, positions 1 and 4 can be thought of as the ‘far’ locations for the inlet and outlet, respectively.
For the inlet, Listing 4, the head HFar has a value of 3 m, as this is the head of the upstream tank,
2in Figure 3, when the system initially is in balance. A flowRate of 50 ×10−3m3/s and a free surface
area Ar of 1.27 m2is specified. This allows the free surface of the upstream tank to vary when the valve
is not fully open. The hydraulic diameter of the patch, dP, has the value 0.222 m and it is the same
everywhere in the system, which is why this value is also seen as the first entry of all the loss factors.
The frictionLossFactors at the inlet has a hydraulic diameter of 0.222 m and an estimated surface
roughness of 0.0025 mm since the pipe is made out of plexiglass. Because of that the computational
domain is 1D, no wall friction is included in the computational domain itself. Instead, the full length of
the pipe is modelled with the boundary condition. The inlet of the computational domain is positioned
at the location of the pressure probe from the lab (3.245 m upstream of the downstream tank), such
10 JONATHAN FAHLBECK∗, H˚
AKAN NILSSON AND SAEED SALEHI
that the time-varying value of the pressure at that patch can be used for the validation. Thus, the
frictionLossFactors of the inlet patch utilises a pipe length of 6.755 m, which is the remainder of the
full length of the 10 m pipe. There are two minor loss factors, minorLossFactors, supplied at the inlet,
both with the hydraulic diameter of 0.222 m. The first entry, expData, has a minor loss coefficient of
45.9 and includes the bend and the upstream valve, 3in Figure 3. It was reported from the lab that
the upstream valve was partly closed, hence the large value of the expData minor loss coefficient. In the
work by Petit et al. [6] a combined minor loss coefficient of 49 was reported for the full system. The
minor loss coefficient of 45.9 for the expData is 49 subtracted by the estimated sum of all the other losses
combined (minor and friction). This is to show that the implementation of the frictionLossFactors
and minorLossFactors work as intended. The second entry, entrance, is due to entrance effects from
the upstream tank to the pipe. A value of 0.45 is assigned in accordance with White [5]. It is assumed
that the friction losses in the upstream tank are negligible, since the velocity in the tank is comparatively
very small.
For the outlet, Listing 5, the same atmospheric pressure pFar is used as for the inlet. The head of the
downstream tank, 6in Figure 3, was at the experimental tests 0.5 m, and the same value is thus given
as the head HFar in the numerical simulation. As mentioned, the same hydraulic diameter dP is used
everywhere. The dynamic minor loss coefficient kDynamic is specified with a table file according to the
time-varying losses in the valve closing and opening, as explained in Section 4.2. The entry dkDynamic
is the hydraulic diameter of the dynamic minor loss coefficient. The frictionLossFactors has a third
entry of 3.245 m, which indicates that the outlet patch is located 3.245 m upstream of the downstream
tank. Thus, from a hydraulic loss perspective, the inlet and outlet of the computational domain are
located at the same position. This is in accordance with the fact that the 1D computational domain
introduced no losses, and that all the losses of the 10 m pipe are given by the boundary conditions. The
hydraulic diameter and surface roughness of the pipe downstream the outlet are the same as for the inlet
(first and second entry, respectively). The final entry is the minorLossFactors, where only one minor
loss factor is specified and it is due to exit effects, from the pipe to the downstream tank. The value of
the exit is according to White [5]. The friction losses in the downstream tank are assumed negligible.
Note that the variable dFar is specified at neither the inlet nor the outlet. By not specifying this, it is
assumed that ‘far’ is located at the surface of a large tank/reservoir, where the velocity is small compared
to that in the computational domain.
For the velocity boundary condition, the pressureInletOutletVelocity is used at both the inlet and
the outlet. All sides use the empty boundary condition.
inlet
{
type headLossPressure;
pFar uniform 0; // m2/s2
HFar 3; // m
flowRate 50e-3; // m3/s
Ar 1.27; // m2
dP 0.222; // m
frictionLossFactors
(
((0.222 0.0025e-3 6.755) pipe)
); // (m, m, m)
minorLossFactor
(
((0.222 45.9) expData)
((0.222 0.45) entrance)
); // (m, -)
}
Listing 4. Inlet patch pressure
boundary condition
outlet
{
type headLossPressure;
pFar uniform 0; // m2/s2
HFar 0.5; // m
dP 0.222; // m
kDynamic tableFile;
kDynamicCoeffs
{
file " FOAM_CASE/constant/kValve";
}// (s, -)
dkDynamic 0.222; // m
frictionLossFactors
(
((0.222 0.0025e-3 3.245) pipe)
); // (m, m, m)
minorLossFactors
(
((0.222 1) exit)
); // (m, -)
}
Listing 5. Outlet patch pressure
boundary condition
5. Results and discussion
Figure 6 shows the presently computed results compared to the numerical work done by Wang et al. [7]
and the experimental test data (only for pressure) [6]. The flow rate is in this work extracted from the
HEAD LOSS PRESSURE BOUNDARY CONDITION 11
volumetric face flux, phi, at the inlet patch, and the pressure probe is located at the centre of the inlet
patch to match the location of the experiment. Neglecting the oscillations that are due to compressibility
effects (not included in the present work), the main variations for both flow rate and static pressure
are greatly captured by the boundary condition. However, a small offset is noted as the valve is fully
closed, as shown in the zoomed view of Figure 6a. This is because the valve could not be fully closed
with the present boundary condition since it is controlled with a loss coefficient and not with a physical
valve. As the valve is closing (5–10 s), the flow rate decreases, and the pressure increases. When the
valve is fully closed (10–18 s), the flow rate is constant at a small value while the pressure continues to
increase linearly. This is due to the fact that the free surface of the upstream tank is rising when the flow
through the pipe is lower than the flow entering the upstream tank. During the valve opening (18–23 s),
the flow rate increases rapidly and the pressure decreases. The static pressure reaches its initial value
fast, while the flow rate requires a longer time to develop, roughly 300 s (not shown here). All these
features are captured adequately by the developed boundary condition. As previously mentioned, the
current implementation cannot capture the oscillations in pressure when the valve is fully closed. This is
because the boundary condition is developed and used for purely incompressible flows. The work done by
Wang et al. [7] concerned water hammer effects, i.e. pressure pulsations in the system, and thus employed
appropriate solver and boundary conditions for such simulations.
(a) Flow rate (b) Pressure probe
Figure 6. Flow rate and static pressure during the transient sequence. Wang et al. [7] is
numerical results with MOC and Exp. is experimental results reported by Petit et al. [6].
As mentioned in Section 4.2, the valve opening and closing sequences are smoothed when the valve
opening is near the fully closed position (v= 2 mm). Without the smoothing, the pressure curve of
Figure 6b shows nonphysical oscillating behaviour at 10 s and 18 s. However, with the smoothing of the
operational sequence, the boundary condition gives a smooth pressure variation, as shown by the zoomed
views of Figure 6b.
The computational domain can easily be extended to 2D or 3D, with resolved boundary layers. Such
results are not shown here, but they present equally convincing results as the 1D results. It can thus be
concluded that the headLossPressure pressure boundary condition can be used to predict a trustworthy
flow rate and pressure at the boundary patches of a computational domain that is connected to a system
of sequential heads and head losses. The feasibility of the developed boundary condition when the flow
field is non-uniform at an inlet patch, or if reverse/mixed flow is encountered at boundary patches needs
further studies.
6. Conclusions
This paper has described and validated a newly developed pressure boundary condition for the open-
source CFD toolbox OpenFOAM, named headLossPressure. The boundary condition is an extended
version of the available totalPressure boundary condition. It adds the possibility to set flow-driving
pressure differences and free surface elevations, and an arbitrary number of hydraulic losses that modify
the pressure at the patches that the boundary condition is used on. This provides the possibility to
model a hydraulic system to which the numerically analysed component is in reality part of. One of the
main motivations behind the implementation is to be able to set boundary conditions for simulations
of hydraulic systems with properties that are known. An example is the head between two free water
surfaces (which can vary in time), which is subjected to flow-dependent (and time-dependent) losses in
the system in which the studied component is located. The total losses are given by the sum of minor
12 JONATHAN FAHLBECK∗, H˚
AKAN NILSSON AND SAEED SALEHI
losses (local in the flow path) and friction losses (occurring in pipes), and the user can supply any number
of losses that are in series in a system. The boundary condition is validated against experimental data,
and the numerical predictions match the relevant features of the experimental to a great extent. The top
features with the boundary condition are:
•Estimates the correct flow rate and pressure at any given operating condition, provided that the
system is well represented by loss factors.
•Models effects, e.g. pressure rise, from a full system to the computational domain in a cost-
efficient manner.
•Models changes in the system, e.g. closing of a valve or head rise via Function1 variables, or
filling/emptying of a reservoir.
•Enables the user to specify the head and/or pressure far from the patch.
•Possible to use for both steady and unsteady computations.
•Compiles directly in OpenFOAM without additional dependencies.
The main limitation with the current development is that it is based on an incompressible assumption,
and that it can therefore not capture pressure waves such as water hammer. It is thus mostly useful for
studies of hydraulic components in a system with incompressible dynamics, or in incompressible systems
where the boundary conditions are unknown at the computational domain boundaries.
Author contributions: Conceptualization, J.F., H.N. and S.S.; methodology, J.F.; software, J.F.; val-
idation, J.F.; formal analysis, J.F.; investigation, J.F.; resources, H.N.; data curation, n/a; writing—
original draft preparation, J.F.; writing—review and editing, J.F., H.N. and S.S.; visualization, J.F.;
supervision, H.N. and S.S.; pro ject administration, J.F.; funding acquisition, H.N. All authors have
read and agreed to the published version of the manuscript. Please turn to the CRediT taxonomy
(https://casrai.org/credit/, accessed on 11 January 2022) for term explanation.
Acknowledgement: This project has received funding from the European Union’s Horizon 2020 re-
search and innovation programme under grant agreement No. 883553. The computations were enabled
by resources provided by the Swedish National Infrastructure for Computing (SNIC) at NSC and C3SE,
partially funded by the Swedish Research Council through grant agreement No. 2018-05973.
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