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Creating Global Citizens with Islamic Values
Turbulence Modelling
Dr. Md. Rezwanul Karim
Associate Professor
Department of Mechanical and Chemical Engineering (MPE)
Islamic University of Technology (IUT)
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Introduction
▪ All flows encountered in engineering practice,
simple ones, such as two-dimensional jets, wakes,
pipe flows and flat plate boundary layers, and
more complicated three-dimensional ones,
become unstable above a certain Reynolds
number.
▪ At low Reynolds numbers, flows are laminar. At
higher Reynolds numbers flows are observed to
become Turbulent.
▪ Turbulence: A chaotic and random state of
motion develops in which the velocity and
pressure change continuously with time within
substantial regions of flow.
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What is Turbulence
▪ The Navier-Stokes equations (for an
incompressible fluid) in a non-dimensional form
contains one parameter: the Reynolds number.
▪ The Reynolds number of a flow gives a measure
of the relative importance of inertia forces
(associated with convective effects) and viscous
forces.
▪ In experiments on fluid systems it is observed that
at values below the so-called critical Reynolds
number (Recrit) the flow is smooth and adjacent
layers of fluid slide past each other in an orderly
fashion.
▪ If the applied boundary conditions do not change
with time the flow is steady. This regime is called
Laminar Flow.
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What is Turbulence
▪ At values of the Reynolds number above Recrit a
complicated series of events takes place which
eventually leads to a radical change of the flow
character.
▪ In the final state, the flow behavior is random
and chaotic. The motion becomes intrinsically
unsteady even with constant imposed
boundary conditions.
▪ The velocity and all other flow properties vary
in a random and chaotic way. This regime is
called Turbulent flow.
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How Turbulence occur?
▪ Small disturbances in the fluid streamlines of a laminar flow can lead to a chaotic and random
state of motion—a turbulent condition.
▪ These disturbances may originate from the free stream of the fluid motion or induced by the
surface roughness where they may be amplified in the direction of the flow, in which case
turbulence will occur.
▪ At low Reynolds number, inertia forces are smaller than the viscous forces. The naturally
occurring disturbances are dissipated away, and the flow remains laminar.
▪ At high Reynolds number, the inertia forces are sufficiently large to amplify the disturbances,
and a transition to turbulence occurs.
▪ Here, the motion becomes intrinsically unstable even with constant imposed boundary
conditions. The velocity and all other flow properties are varying in a random and chaotic
way.
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How Turbulence occur?
▪ A typical point velocity measurement might exhibit the form
shown in the figure.
▪ The dependent variable such as the transient velocity distribution
in Fig.1 must be interpreted as an instantaneous velocity.
▪ The random nature of a turbulent flow precludes an economical
description of the motion of all the fluid particles.
▪ Instead, the velocity in the figure is decomposed into a steady
mean value U with a fluctuating component u′(t) superimposed
on it: u(t) = U + u′(t). This is called the Reynolds decomposition.
▪ A turbulent flow can now be characterised in terms of the mean
values of flow properties (U, V, W, P etc.) and some statistical
properties of their fluctuations (u′, v′, w′, p′ etc.). It is most
attractive to characterize a turbulent flow by the mean values of
flow properties with its corresponding statistical fluctuating
property.
Fig.1: Typical point velocity measurement in
turbulent flow
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Energy Cascade
▪ Even in flows where the mean velocities and pressures vary
in only one or two space dimensions, turbulent fluctuations
always have a three-dimensional spatial character.
▪ Furthermore, visualizations of turbulent flows reveal
rotational flow structures, so-called turbulent eddies, with
a wide range of length and velocity scales called
turbulent scales.
▪ The figure depicts a cross-sectional view of a turbulent
boundary layer on a flat plate, shows eddies whose length
scale is comparable with that of the flow boundaries as
well as eddies of intermediate and small size.
▪ The largest eddies have a characteristic velocity and a
characteristic length of the same order as the velocity and
length scale of the mean flow.
▪ The large eddies are therefore effectively inviscid due to
fact that inertia effects dominate viscous effects.
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Energy Cascade
▪ The largest turbulent eddies interact with and extract energy
from the mean flow by a process called vortex stretching.
▪ Vortex stretching is a phenomenon in turbulent flow where
the rotational structures, known as vortices, are extended or
stretched in the direction of the flow due to the intense
velocity gradients and shearing present in turbulent fluid
motion. Suitably aligned eddies are stretched because one
end is forced to move faster than the other.
▪ During vortex stretching, the angular momentum is
conserved, and the stretching work done by the mean flow on
the large eddies provides the energy that maintains the
turbulence.
▪ These larger eddies then breed new instabilities creating
smaller eddies that are transported mainly by vortex
stretching from the larger eddy rather than from the mean
flow.
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Energy Cascade
▪ In this way the kinetic energy is handed down from
large eddies to progressively smaller and smaller
eddies in what is termed the energy cascade. This
process continues until the eddies become so small that
viscous effects become important (the eddy length
scales ul≪ν).
▪ Larger eddies are flow-dependent as they are
generated from mean flow characteristics; thus, their
turbulent scales are large compared with viscosity
causing the structure of the eddy to be highly
anisotropic.
▪ Small eddies have much smaller turbulent scales (with
scales up to the order of 10-4) compared with viscosity
causing the flow to be isotropic since the diffusive
effects of viscosity dominate and smear out the
directionality of the flow structure.
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Examples of simple Turbulent flows
▪ Some examples of simple turbulent flow are a jet entering a domain with stagnant fluid, a
mixing layer and the wake behind objects such as cylinders.
▪ Such flows are often used as test cases to validate the ability of computational fluid dynamics
software to accurately predict fluid flows.
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Transition from Laminar to
Turbulent flow
▪ The initial cause of the transition to turbulence can be explained by considering the
stability of laminar flows to small disturbances.
▪ Small disturbances in the fluid streamlines of a laminar flow can lead to a chaotic and
random state of motion—a turbulent condition.
▪ These disturbances may originate from the free stream of the fluid motion or induced by
the surface roughness where they may be amplified in the direction of the flow, in which
case turbulence will occur.
▪ At low Reynolds number, inertia forces are smaller than the viscous forces. The naturally
occurring disturbances are dissipated away, and the flow remains laminar.
▪ At high Reynolds number, the inertia forces are sufficiently large to amplify the
disturbances, and a transition to turbulence occurs.
▪ Here, the motion becomes intrinsically unstable even with constant imposed boundary
conditions. The velocity and all other flow properties are varying in a random and chaotic
way.
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Transition from Laminar to
Turbulent flow
▪ The photographs show the flow in a boundary layer.
▪ Below Recrit the flow is laminar and adjacent fluid layers slide past
each other in an orderly fashion.
▪ The flow is stable. Viscous effects lead to small disturbances being
dissipated.
▪ Above the transition point, Recrit small disturbances in the flow start
to grow.
▪ A complicated series of events takes place that eventually leads to
the flow becoming fully turbulent.
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Transition processes in boundary
layer flow over a flat plate
▪ The developing flow over a flat plate is such a flow,
and the transition process has been extensively
researched for this case.
▪ If the incoming flow is laminar numerous
experiments confirm the predictions of the theory
that initial linear instability occurs around Rex,crit =
91 000.
▪ The unstable two-dimensional disturbances are
called Tollmien-Schlichting (T–S) waves. These
disturbances are amplified in the flow direction.
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Transition processes in boundary
layer flow over a flat plate
▪ The subsequent development depends on the
amplitude of the waves at maximum (linear)
amplification. Since amplification takes place over
a limited range of Reynolds numbers, it is possible
that the amplified waves are attenuated further
downstream and that the flow remains laminar.
▪ If the amplitude is large enough a secondary, non-
linear, instability mechanism causes the Tollmien–
Schlichting waves to become three-dimensional
and finally evolve into hairpin Λ-vortices.
▪ In the most common mechanism of transition, so-
called K-type transition, the hairpin vortices are
aligned.
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Transition processes in boundary
layer flow over a flat plate
▪ Above the hairpin vortices a high shear region is induced
which subsequently intensifies, elongates and rolls up.
Further stages of the transition process involve a cascading
breakdown of the high shear layer into smaller units with
frequency spectra of measurable flow parameters
approaching randomness.
▪ Regions of intense and highly localised changes occur at
random times and locations near the solid wall. Triangular
turbulent spots burst from these locations.
▪ These turbulent spots are carried along with the flow and
grow by spreading sideways, which causes increasing
amounts of laminar fluid to take part in the turbulent motion.
▪ Transition of a natural flat plate boundary layer involves the
formation of turbulent spots at active sites and the
subsequent merging of different turbulent spots convected
downstream by the flow. This takes place at Reynolds
numbers Rex,tr ≈ 106.
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Why turbulent flows are
challenging?
▪ Unsteady aperiodic motion
▪ Fluid properties exhibit random spatial variations (3D)
▪ Strong dependence from initial conditions
▪ Contain a wide range of scales (eddies)
▪ The implication is that the turbulent simulation MUST be always
three-dimensional, time accurate with extremely fine grids
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Turbulence models
• A turbulence model is a computational procedure to close the system of mean flow
equations.
• For most engineering applications it is unnecessary to resolve the details of the turbulent
fluctuations.
• Turbulence models allow the calculation of the mean flow without first calculating the full
time-dependent flow field.
• We only need to know how turbulence affected the mean flow.
• In particular we need expressions for the Reynolds stresses.
• For a turbulence model to be useful it:
– must have wide applicability,
– be accurate,
– simple,
– and economical to run.
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Closure problem
▪ The Navier–Stokes equations govern the velocity and pressure of a fluid flow.
▪ In a turbulent flow, each of these quantities may be decomposed into a mean part and
a fluctuating part.
▪ Averaging the equations gives the Reynolds-averaged Navier–Stokes (RANS)
equations, which govern the mean flow.
▪ The nonlinearity of the Navier–Stokes equations means that the velocity fluctuations
still appear in the RANS equations, in the nonlinear term −𝜌(𝑣_𝑖^′ 𝑣_𝑗^′ ) ̅ from the
convective acceleration.
▪ This term is known as the Reynolds stress, Rij Its effect on the mean flow is like that of a
stress term, such as from pressure or viscosity.
▪ To obtain equations containing only the mean velocity and pressure, we need to close
the RANS equations by modelling the Reynolds stress term , Rij, as a function of the
mean flow, this is the closure problem.
