Basics of Turbulent Flow

Basics of Turbulent Flow
Khusro Kamaluddin

Introduction
It is the natural state of most of the fluids.
It is the underlying principle in:
Cloud Formation
Atmospheric Transport
Practical Droplet
Spray
Combustion Process
Wind Harvesting
Industrial Particle Production and Transport

I. Irregular/Random
It is defined by its irregularity and randomness.
It is unsteady and fluctuates randomly throughout the domain of
space and time.
It has random velocity fluctuations with a wide range of length
and time scales.

II. Rich in Scales of Eddy Motion
A single turbulent flow will have a wide variation of length and
time scale.
Large scales of motion are:
➢ Influenced by geometry of flow.(Boundary Conditions)
Small scales of motion are:
➢ Influenced by rate of energy transfer from large scales
➢ Influenced by Viscosity of the fluid
➢ Universal in character
➢ Independent of flow geometry

II. Rich in Scales of Eddy Motion
❑ Different size of eddy have corresponding Kinetic Energies
❑ Can be determined by:
➢ Vorticity
ഥ
Ω = ∇ × ഥ
U
➢ Intensity of velocity fluctuation
1
2
𝑚𝑢2
❑ Average distribution of energy between frequency is called
energy spectrum.

III. Large Reynolds Number
𝑅𝑒 =
𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝐹𝑜𝑟𝑐𝑒
𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝐹𝑜𝑟𝑐𝑒
=
𝑈𝐿
𝜈
❑ For The same 𝑈&𝐿
❑ For Laminar Flow Kinematic Viscosity if High
❑ Thus suppressing the fluctuations
❑ The energy is dissipated via molecular diffusion.

IV. Dissipative
❑ Viscosity removes flow fluctuations by converting the associated
Kinetic Energy into heat.
❑ Flow remains laminar when small perturbations are damped out
by viscosity.
❑ Re(↑) → Viscous Forces(↓)
❑ Damping derived from molecular diffusion of momentum is
unable to dissipate large perturbations.
❑ Turbulence often originates from instabilities in laminar
flow.(Related to interaction of viscous term and non-linear inertia
term in equation of motion)

IV. Dissipative
❑ Perturbations can occur from:
➢ Slight thermal current
➢ Surface Roughness
➢ Microscopic sources(Subcontinuum molecular motion like
Brownian Motion)

V. Highly Vortical
❑ It is rotational in nature.
❑ Characterized by high level of fluctuating Vorticity.
❑ Velocity derivatives are dominated by smallest scales of
turbulence.

VI. 3-Dimentional
❑ It is a 3-Dimentional phenomenon
❑ 2D Turbulence is used to describe simple cases where flow is
restricted to two dimensions.
❑ 2D Turbulence is not true Turbulence.
❑ Vorticity fluctuations cannot be 2-Dimentional.
❑ As “Vortex Stretching”, an important vorticity maintenance
mechanism is not present in 2-Dimentional flow.

VII. Highly Diffusive
❑ Its diffusivity is much higher than of laminar flow(Molecular
Diffusivity)
❑ This causes:
➢ Rapid Mixing
➢ Increased rate of momentum transfer
➢ Increased rate of heat transfer
➢ Increased rate of mass transfer

VIII. Not a Fluid Behavior
❑ It is a flow feature, not a fluid property.
❑ If is different for different flows.
➢ But some characteristics remain common.

IX. Holds Continuum Hypothesis
❑ Turbulence is a continuum phenomenon.
❑ It is governed by the equations of fluid mechanics.
❑ The smallest turbulent length scales are larger than the
molecular mean free path.
❑ An exception to this is Rarified Gas Medium.

I. Leonardo Da Vinci(1452-1519)
A deluge, c.1517–18, Pen and black ink with wash on paper, RCIN 912380 via Google Art Project. Wikimedia Commons
Studies of water passing obstacles and falling, c. 1508-9. Wikimedia Commons

II. Lord Rayleigh(1842-1919)
❑ Worked on stability of parallel flow inviscid fluid.
❑ Found that presence of an inflection point was a necessary and
sufficient condition for an inviscid fluid to become unstable.
➢ Inflection point signifies that some sort of flow deceleration is
required for inviscid flow to become unstable.
❑ Suggested that presence of viscosity can promote initiation of
turbulence by enabling the creation of inflection point.

II. Lord Rayleigh(1842-1919)
❑ Indicated that viscosity is a double edged sword
➢ It is needed to kill turbulence.
➢ But simultaneously creates it.

III. Osborne Reynolds(1842-1912)
❑ Used 6ft long glass tube Ø2.63, 1.53 & 0.789 cm with Trumpet
mouths
❑ Flow is laminar at low Reynolds Number where perturbations are
damped out by viscosity.

❑ At high flow rate the color band appears to expand and mix with
water.
❑ On viewing under electric spark the mass of the color resolves
into less distinct curls, in which eddies can be seen.

❑ In transition region he saw intermittent character of fluid motion
caused by disturbance, which appears as flashes succeeding each
other.
❑ He saw sporadic* bursts of turbulence alternating with laminar
flow.
*(Occurring at irregular intervals or only in a few places, scattered & isolated)

❑ Poiseuille flow is stable to infinitesimal perturbations at all
Reynolds Number values.
❑ This implies that transition to turbulence is dependent on
perturbations, in addition to Reynolds Number.
❑ This is unlike flow over buff bodies, where instabilities can be
predicted using linearized perturbation theory with an
infinitesimal amplitude.
❑ Thus indicating presence of more than one way to create a
turbulent flow.

❑ For practical pipe flows
❑ Transition occurs at Re~2000
❑ Fully Turbulent flow occurs at Re~2300
❑ These Reynolds Numbers are difficult to predict.
❑ They depend on:
❑ Upstream Flow Conditions.
❑ Boundary Roughness/Texture.
❑ With sufficient care, laminar flow in a pipe can be maintained to
atleast Re~105

❑ Vortical structures are initially 2D
❑ Generated due to shear(no
slip) in boundary layer
❑ These vortical structures become
progressively unstable farther
downstream
❑ 3D interactions eventually
leading to turbulence

IV. Henri Bernard(1914-1942)
❑ Fluid is confined between 2 large plates
❑ Heat is conducted through the quiescent fluid via molecular
diffusion/conduction.
❑ Upward buoyant force if balanced by vertical pressure
gradient
❑ When critical temperature difference is reached
❑ Laminar Bernard cells are formed

IV. Henri Bernard(1914-1942)
❑ Instability criterion is
𝑅𝑎 ≡
𝐺𝑟𝑎𝑣𝑖𝑡𝑦
𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝐷𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦
=
𝑔𝛽∆𝑇ℎ3
𝜈𝛼
= 1.70 × 103

VI. Ludwig Prandtl(1875-1953)
❑ Introduced the concept of “Mixing Length”(Inline with the Kinetic
Theory of Gases )
❑ Mixing length is the average distance that a fluid element
would stray from the mean streamline.
❑ Turbulence is not a feature of the fluid but the flow
❑ Thus the anlogy between “Mean free path” and “Mixing Length”

Continuum Hypothesis
❑ For continuum hypothesis to be valid the mean free path of the
molecule should be very small relative to system geometry.
𝐾𝑛𝑢𝑑𝑠𝑒𝑛 𝑁𝑜. (𝐾𝑛) ≡
𝜆𝑚𝑓𝑝
𝑙
❑ 𝜆𝑚𝑓𝑝 is the mean free path of the molecule
❑ 𝑙 is the smallest geometric scale of the flow
𝜆𝑚𝑓𝑝 < 𝑙𝑚𝑒𝑑𝑖𝑢𝑚 < 𝑙
❑ Under this condition the continuum fluid properties can be
averaged over the volume of size

Continuum Hypothesis
x
y
z
𝒅𝒙
𝒅𝒚
𝝀𝒎𝒆𝒅𝒊𝒖𝒎
𝝀
𝒎𝒆𝒅𝒊𝒖𝒎

Eulerian Frame
❑ Implemented by fixing control volume in space.
❑ Monitoring the flow passing through the CV over a period of
time.
❑ The independent variables are spatial coordinate 𝑥, 𝑦, 𝑧 and
the time 𝑡 .
❑ The properties in a flow field are specified in terms of space
coordinates and time.

Lagrangian Frame
❑ This approach keeps track of a particular mass of fluid.
❑ Consider a case where fluid particle is initially(at time to) is at
position 𝑋+ 𝑡𝑜, ത
𝑌 = ത
𝑌.
❑ ത
𝑌 is lagrangian or material coordinate.
❑ The particle moves with local fluid velocity
𝜕𝑋+ 𝑡𝑜, ത
𝑌
𝜕𝑡
= ഥ
𝑈 𝑋+ 𝑡𝑜, ത
𝑌 , 𝑡

Eulerian Frame
www.youtube.com. (n.d.). YouTube. [online] Available at:https://www.youtube.com/watch?v=IxOt4ALui8k.

Lagrangian Frame
www.youtube.com. (n.d.). YouTube. [online] Available at:https://www.youtube.com/watch?v=SsnYoqdFGIY&t=1s.

Eulerian-Lagrangian Transformation

Mass Conservation
x
y
z
𝒅𝒙
𝒅𝒚
o
❑ Density is 𝜌 at point o.
❑ Velocity field at point o is given by
ഥ
𝑈 = 𝑈 Ƹ
𝑖 + 𝑉 Ƹ
𝑗 + 𝑊෠
𝑘
❑ Invoking Taylor series expansion about
point o yields:
𝜌𝑥+
𝑥
2
= 𝜌 +
𝜕𝜌
𝜕𝑥
𝑑𝑥
2
+
𝜕2𝜌
𝜕𝑥2
1
2!
𝑑𝑥
2
2
+ ⋯
❑ Neglecting higher terms yield:
𝜌𝑥+
𝑥
2
= 𝜌 +
𝜕𝜌
𝜕𝑥
𝑑𝑥
2
& similarly 𝑈𝑥+
𝑥
2
= U +
𝜕𝑈
𝜕𝑥
𝑑𝑥
2
❑ Where 𝜌, 𝑈,
𝜕𝜌
𝜕𝑥
&
𝜕𝑈
𝜕𝑥
are evaluated at o.

Mass Conservation
x
y
z
𝒅𝒙
𝒅𝒚
o
❑ Mass flux for each of the 6 surfaces of the
CV can be described as:
‫׬‬
𝐶𝑆
𝜌ഥ
𝑈. 𝑑 ҧ
𝐴
❑ ሶ
𝑚𝑖𝑛 − ሶ
𝑚𝑜𝑢𝑡 =
𝜕
𝜕𝑡
(𝑚𝑒𝑙𝑒𝑚𝑒𝑛𝑡)
❑ For the (+x) face we have:
❑ 𝜌 +
𝜕𝜌
𝜕𝑥
𝑑𝑥
2
U +
𝜕𝑈
𝜕𝑥
𝑑𝑥
2
𝑑𝑦𝑑𝑥
❑ 𝜌U + 𝜌
𝜕𝜌
𝜕𝑥
𝑑𝑥
2
+ U
𝜕𝑈
𝜕𝑥
𝑑𝑥
2
+
𝜕𝜌
𝜕𝑥
𝜕𝑈
𝜕𝑥
𝑑𝑥
2
2
𝑑𝑦𝑑𝑧
❑ 𝜌U + 𝜌
𝜕𝜌
𝜕𝑥
𝑑𝑥
2
+ U
𝜕𝑈
𝜕𝑥
𝑑𝑥
2
𝑑𝑦𝑑𝑧

Mass Conservation
x
y
z
𝒅𝒙
𝒅𝒚
o
❑ 𝜌U + 𝜌
𝜕𝜌
𝜕𝑥
𝑑𝑥
2
+ U
𝜕𝑈
𝜕𝑥
𝑑𝑥
2
𝑑𝑦𝑑𝑧
❑ 𝜌Udxdydz +
𝜌
2
𝜕𝜌
𝜕𝑥
dxdydz +
𝑈
2
𝜕𝑈
𝜕𝑥
dxdydz
❑ +x: +𝜌Udxdydz +
1
2
𝜕(𝜌𝑈)
𝜕𝑥
dxdydz
❑ -x: −𝜌Udxdydz +
1
2
𝜕(𝜌𝑈)
𝜕𝑥
dxdydz
❑ +y: +𝜌Vdxdydz +
1
2
𝜕(𝜌𝑉)
𝜕𝑦
dxdydz
❑ -y: −𝜌Vdxdydz +
1
2
𝜕 𝜌𝑉
𝜕𝑦
dxdydz
❑ +z: +𝜌Wdxdydz +
1
2
𝜕(𝜌𝑊)
𝜕𝑧
dxdydz
❑ -z: −𝜌Wdxdydz +
1
2
𝜕 𝜌𝑊
𝜕𝑧
dxdydz
❑
𝜕(𝜌𝑈)
𝜕𝑥
+
𝜕(𝜌𝑉)
𝜕𝑦
+
𝜕(𝜌𝑊)
𝜕𝑧
dxdydz ≡ 𝑁𝑒𝑡 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑀𝑎𝑠𝑠 𝐹𝑙𝑢𝑥 𝑇ℎ𝑟𝑜𝑢𝑔ℎ 𝐶𝑉
𝜕(𝜌𝑈)
𝜕𝑥
dxdydz
𝜕(𝜌𝑉)
𝜕𝑦
dxdydz
𝜕(𝜌𝑊)
𝜕𝑧
dxdydz
+
+

Mass Conservation
x
y
z
𝒅𝒙
𝒅𝒚
❑ ሶ
𝑚𝑜𝑢𝑡 − ሶ
𝑚𝑖𝑛 = +
𝜕(𝜌𝑈)
𝜕𝑥
+
𝜕(𝜌𝑉)
𝜕𝑦
+
𝜕(𝜌𝑊)
𝜕𝑧
dxdydz
❑ ሶ
𝑚𝑖𝑛 − ሶ
𝑚𝑜𝑢𝑡 = −
𝜕(𝜌𝑈)
𝜕𝑥
+
𝜕(𝜌𝑉)
𝜕𝑦
+
𝜕(𝜌𝑊)
𝜕𝑧
dxdydz
❑
𝜕
𝜕𝑡
(𝑚𝑒𝑙𝑒𝑚𝑒𝑛𝑡) = −
𝜕(𝜌𝑈)
𝜕𝑥
+
𝜕(𝜌𝑉)
𝜕𝑦
+
𝜕(𝜌𝑊)
𝜕𝑧
dxdydz
❑
𝜕𝜌
𝜕𝑡
𝑑𝑥𝑑𝑦𝑑𝑧 = −
𝜕(𝜌𝑈)
𝜕𝑥
+
𝜕(𝜌𝑉)
𝜕𝑦
+
𝜕(𝜌𝑊)
𝜕𝑧
dxdydz
❑
𝜕𝜌
𝜕𝑡
+
𝜕(𝜌𝑈)
𝜕𝑥
+
𝜕(𝜌𝑉)
𝜕𝑦
+
𝜕(𝜌𝑊)
𝜕𝑧
dxdydz = 0
❑
𝜕𝜌
𝜕𝑡
+
𝜕(𝜌𝑈)
𝜕𝑥
+
𝜕(𝜌𝑉)
𝜕𝑦
+
𝜕(𝜌𝑊)
𝜕𝑧
= 0
❑
𝜕𝜌
𝜕𝑡
+ ∇ ∙ 𝜌ഥ
𝑈 = 0
❑ ∇ ∙ 𝑈 = 0

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