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J. Szantyr – Lecture No. 6 – Fluid – Solid Interaction –
Concept of the Entrained Mass of Fluid
In 1828 Friedrich Bessel has noticed that a pendulum immersed
in water changes (increases) its period of oscillations in
comparison to the value in air. This may be interpreted as a
virtual increase of mass of the pendulum. Bessel has introduced
the idea of the entrained mass of water, i.e. a certain mass of
water (in general: fluid), exercising the unsteady motion
together with the immersed object and changing its motion
characteristics. The entrained mass increases the inertia of the
object, introducing additional forces to the description of
Friedrich Wilhelm Bessel
1784 - 1846
motion.
The additional forces on the object exercising an
accelerated motion in a real, viscous fluid (in contrast to
the motion in a vacuum) may be divided into two parts: a
part associated with acceleration of a certain mass of
fluid (in principle a completely potential effect) and a
part resulting from viscous effects in the unsteady
boundary layer forming on the solid object. This second
Alfred Barnard Basset
part is called Basset force (1888).
1854 - 1930
The Basset force is important first of all for small solid objects moving in a
fluid. Its magnitude depends on the history of motion and in the case of a
spherical object it may be described by the following formula:
F (t) =
where:
t
3 2
D πρC µC ∫
2
0
Du Dv
−
Dt ′ Dt ′ dt ′
t − t′
D – object diameter
t – current time
ρ C - fluid density
µC - fluid dynamic viscosity coefficient
u - velocity of the object
v
- velocity of the fluid
In the physical sense the Basset force results from a retarded formation of
the boundary layer and viscous wake behind the solid object moving in the
fluid in an accelerated way.
The simplest interpretation: entrained mass determines the work required to
change the kinetic energy of the fluid due to an accelerated motion of the
immersed solid object. The kinetic energy of the fluid motion caused by the
moving solid object may be written as:
ρ
where V – the entire fluid volume
E=
u12 + u22 + u32 dV
(
∫
2
)
V
2
In a steady linear motion there is E=const and E ∝ U
 u1  2  u2  2  u3  2 
ρ
2
E = IU where: I = ∫   +   +   dV
2
V 
 U   U   U  
Then we may write:
If the object accelerates or brakes then the energy E changes with the velocity U.
The change of energy E may be caused only by the work of an additional
hydrodynamic force F, which appears on the object in an unsteady motion,
according to the relation:
force F is similar to the force required to
1 dE
dU
F =−
= − ρI
accelerate the object of mass m i.e.: m dU dt
U dt
dt
It is convenient to describe the force F as an additional mass of fluid M=ρI
accelerated together with the object. In reality every fluid particle around
the object experiences different acceleration, hence the entrained mass M is
a certain „virtual mass.”.
A simple example – linear accelerated motion of a sphere or a
cylinder in a two-dimensional flow (2D):
The potential fluid flow
description is applied
Stream lines and
equipotential lines
Velocity vectors and
pressure field
The following velocity potentials are obtained:
For the sphere:
UR 3
Φ ( r , ϑ ) = − 2 cos ϑ
2r
UR 2
For the cylinder: Φ ( r ,ϑ ) = −
cos ϑ
r
Then the integrals I determining the entrained mass may be calculated as:
For the sphere:
 1 ∂Φ  2  1 ∂Φ  2 
2 3
2
I = ∫ ∫ 
 +
  2πr sin ϑdϑdr = πR
3
R 0 
 U ∂r   Ur ∂ϑ  
∞ 2π
i.e. it is equal to
half of the fluid
mass displaced
by the sphere
For the cylinder (per unit length):
 1 ∂Φ  2  1 ∂Φ  2 
2
I = ∫ ∫ 
 +
 rdϑdr = πR
R 0 
 U ∂r   Ur ∂ϑ  
∞ 2π
i.e. it is equal to
the fluid mass
displaced by the
cylinder
In the general case of motion of an object in six degrees of freedom the
unsteadiness of any velocity component results in generation of additional forces
in all six degrees of freedom. Then we obtain a matrix (a tensor) of entrained
masses: M ij
Fi = − M ij u j
i,j=1,2,3,4,5,6
It may be shown that in a potential flow the matrix of entrained masses is
symmetrical, hence in a general case we may have 21 independent entrained
masses. Symmetry of the moving solid object may lead to further reduction of
the number of entrained masses.
Tensor of the entrained masses
Symmetry:
M 15
M 16
u 1
M 22 M 13 M 24 M 25
M 26
u 2
M 33 M 34 M 35
M 36
u 3
kg
M 41 M 42 M 43 M 44 M 45 M 46
u 4
kgm
M 51 M 52 M 53 M 54 M 55 M 56
u 5
kgm 2
M 61
u 6
M 11 M 12
M ij = M ji
Dimensions
M 21
F=
M 31 M 32
M 13 M 14
M 62 M 63 M 64 M 65 M 66
First index – direction of force, second index – direction of motion
i=1,2,3 – forces; i=4,5,6 - moments
j=1,2,3 – linear accelerations; j=4,5,6 – angular accelerations
Sometimes the entrained masses are presented in a non-dimensional form,
i.e. related to the respective mass characteristic of the solid object.
Non-dimensional coefficient of the entrained masses are denotedmas
ij
Calculation of the entrained mass coefficients for a three-dimensional object of
an arbitrary geometry is difficult. If one dimension of the object is significantly
larger than others, then the so called slender body theory may be applied.
In this theory the object may be cut into „slices” and the entrained mass
coefficients for two-dimensional sections may be integrated along the object:
m22 = ∫ a22 dx
m23 = − ∫ a23dx
m24 = ∫ a24 dx
m26 = ∫ xa22 dx
m33 = ∫ a33 dx
m35 = − ∫ xa33 dx
m44 = ∫ a44 dx
m46 = ∫ xa24 dx
m55 = ∫ x 2 a33dx
m66 = ∫ x 2 a22 dx
L
L
L
L
L
L
L
L
l
L
In more complicated cases the
commercial CFD software is used
The entrained mass coefficients for some selected
two-dimensional sections are given below:
In an unsteady motion of the solid object immersed in fluid the
entrained mass is a virtual mass of fluid performing motion with
the same velocity as the solid object. The entrained mass
increases the inertia of the object and in this way it influences the
motion characteristics of the object.
In reality the motion of the immersed solid object induces the
motion of another mass of fluid with diverse velocities – higher
velocity close to the object and smaller at larger distances from
it. This real mass of moving fluid increases the inertia of the
object in the same way as the virtual entrained mass.
For objects moving in gases the entrained mass of gas is usually
not taken into account due to the small density of gases.
Influence of the entrained mass on the solid object
oscillations – a simple one-dimensional example.
m – mass of the object
c – damping coefficient (due to fluid viscosity)
k – restoring force coefficient
x – object displacement
The entrained mass increases the inertia of the object, thus it counteracts
oscillations. In this case the equation describing oscillations has the form:
mx + cx + kx = −ma x
( m + ma ) x + cx + kx = 0
where:
ma - entrained mass
me - „effective” mass
me x + cx + kx = 0
The own frequency of oscillations of the
immersed body may be determined as:
1
fn =
2π
k
c2
1−
me
4me k
It should be noticed that immersion of the
oscillating object results in reduction of
the own frequency of oscillations.
Influence of the entrained mass on the vibration of the reversible machine
(pump-turbine) rotor
Model experiments
Model of the rotor of a reversible
machine (pump-turbine) has been tested
in air and in water. Model vibration was
excited by an inducer in 384 points
shown in the picture. The responses for
several own modes of vibration were
registered.
Change of the damping coefficients
Change of own frequencies
Own frequencies and damping coefficients in air and in water
Influence of the entrained mass on the vibration of the pump rotor
Numerical calculations
The basic own
vibration modes
2ND
Model of the rotor
0ND
3ND
Calculations were performed using
the Finite Element Method. The
computational model of the rotor was
built of 165000 quadrihedral
elements, and the model of the
surrounding fluid was built of 342676
such elements.
Comparison of the calculated
(SIM) and measured (EXP)
own vibration frequencies of
the rotor in air and in water
Influence of the entrained mass on the vibration of the ship
propulsion system
Scheme of a ship propulsion system. The most important are the torsional
and longitudinal (axial) vibrations. An important component of the system
is the propeller, being a heavy object immersed in water. Variable
hydrodynamic forces are generated on the propeller, constituting the main
source of vibration excitation.
Variable thrust force
Variable torque on the shaft -->
Determination of the entrained mass for the propeller is necessary for correct
analysis of vibration of the ship propulsion system. There are many methods for
determination of the entrained mass. The simplest are the empirical formulae:
For axial vibration:
M 11 = ( 0,1 − 0,2 ) M P
or:
M P - mass of the propeller
D – propeller diameter
K1 = 0,25 − 0,30
K 2 = 19 − 28
M 11 = C11 ρD 3
For torsional vibration:
M 44
M PD
= K1
K2
or:
M 44 = C44 ρD 5
C11 =
C44 =
( D) A
2
0,2 P
2
z
( D) A
0,0224 P
2
2
z
P – propeller pitch
z – propeller number of blades
A – propeller area coefficient
ρ - density of water
Computational determination of the entrained mass for a Kaplan turbine
Scheme of the Kaplan turbine
Rotor model in the Finite
Element Method
The objective of calculations was to determine the entrained masses for
different rotor sizes, different numbers of blades and different blade pitch
settings. Calculations were performed for turbines having powers from 3
[MW] to 75 [MW] and rotor diameters from od 1.75 [m] to 7.5 [m].
The entrained mass of the rotor in
transverse direction M 22 for the
diameter 4.5 [m] at different blade
pitch settings
The entrained
mass M 22 for
rotors of
different
diameters and
numbers of
blades at the
same pitch
setting angle β
Influence of the entrained mass on vibration of the Francis water turbine
Model experiments
Model of the rotor
Test set-up for measurements in water and in air
The rotor model was excited using a special inducer (hammer), in 118
selected points, exciting different modes of own vibration in air and in water.
The vibrations were measured and registered using special sensoring system.
One degree of freedom vibrations of the rotor may be described by the
following equation:
( M W + M A ) X + ( CW + C A ) X + ( KW + K A ) X = F ( t )
Index A denotes the effect of immersion of the rotor in water, concerning
the mass M, the damping coefficient C and the stiffness coefficient K.
Influence of the entrained mass on vibration of the Francis water turbine
Numerical calculations
Model of the rotor for FEM calculation
Measured and calculated degree
of reduction of frequency of the
different vibration modes in water
Every geometrically repeatable sector of
the rotor was modelled by 6133
hexahedral finite elements. The results of
calculations were compared with
experimental measurements discussed
before. In all cases the immersion in
water has reduced the own frequencies of
vibration of the rotor.
Comparison of the calculated and
measured frequencies of the different
vibration modes in air and in water
1/--pages
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