57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
1
Chapter 3 Bernoulli Equation
3.1 Flow Patterns: Streamlines, Streaklines, Pathlines
A streamline is a line that is everywhere tangent to the velocity field.
Streamlines are obtained analytically by integrating the equations defining lines
tangent to the velocity field. For two-dimensional flows the
slope of the streamline, 𝑑𝑦⁄𝑑𝑥, must be equal to the tangent of the angle that the velocity vector makes with the 𝑥
axis or
𝑑𝑦 𝑣
=
𝑑𝑥 𝑢
If the velocity field is known as a function of 𝑥 and 𝑦 (and 𝑡 if
the flow is unsteady), this equation can be integrated to give
the equation of the streamlines.
A pathline is the line traced out by a given particle as it flows from one
point to another. The pathline is a Lagrangian concept that can be produced in
the laboratory by marking a fluid particle (dying a small fluid element) and taking
a time exposure photograph of its motion.
Illustration of a pathline (left) and an example of pathlines, motion of water induced by surface waves (right).
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
2
A streakline consists of all particles in a flow that have previously passed
through a common point. Streaklines are more of a laboratory tool than an analytical tool. They can be obtained by taking instantaneous photographs of marked
particles that all passed through a given location in the flow field at some earlier
time. Such a line can be produced by continuously injecting marked fluid (neutrally buoyant smoke in air, or dye in water) at a given location.
Illustration of a streakline (left) and an example of streaklines, flow past a full-sized streamlined vehicle in the GM aerodynamics laboratory wind tunnel, and 18-ft by 34-ft test section
facility driven by a 4000-hp, 43-ft-diameter fan (right).
If the flow is steady pathlines, streamlines, and streaklines are the same.
For unsteady flows none of these three types of lines need be the same.
3.2 Streamline Coordinates
In the streamline coordinate system the flow is described in terms of one
coordinate along the streamlines, denoted 𝑠, and the second coordinate normal
to the streamlines, denoted 𝑛. Unit vectors in these two directions are denoted
̂ , as shown in Fig. 4.8. Of the major advantages of using the streamline
by 𝒔̂ and 𝒏
coordinate system is that the velocity is always tangent to the 𝑠 direction. That is,
V = 𝑣𝑠 𝒔̂
This allows simplifications in describing the fluid particle acceleration and in solving the equations governing the flow.
Chapter 3
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Figure 4.8 Streamline coordinate system for two-dimensional flow.
For two-dimensional flow we can determine the acceleration as
𝑎=
𝐷V
̂
= 𝑎𝑠 𝒔̂ + 𝑎𝑛 𝒏
𝐷𝑡
where 𝑎𝑠 and 𝑎𝑛 are the streamline and normal acceleration, respectively. In
general, for steady flow both the speed and the flow direction are a function of
location 𝑉 = 𝑉(𝑠, 𝑛) and 𝒔̂ = 𝒔̂(𝑠, 𝑛). Thus, application of the chain rule gives
𝑎=
𝐷(𝑣𝑠 𝒔̂) 𝐷𝑣𝑠
𝐷𝒔̂
=
𝒔̂ + 𝑣𝑠
𝐷𝑡
𝐷𝑡
𝐷𝑡
or
𝑎=(
𝜕𝑣𝑠 𝜕𝑣𝑠 𝑑𝑠 𝜕𝑣𝑠 𝑑𝑛
𝜕𝒔̂ 𝜕𝒔̂ 𝑑𝑠 𝜕𝒔̂ 𝑑𝑛
+
+
+
) 𝒔̂ + 𝑣𝑠 ( +
)
𝜕𝑡
𝜕𝑠 𝑑𝑡 𝜕𝑛 𝑑𝑡
𝜕𝑡 𝜕𝑠 𝑑𝑡 𝜕𝑛 𝑑𝑡
This can be simplified by using the fact that the velocity along the streamline is
𝑣𝑠 =
𝑑𝑠
𝑑𝑡
and the particle remains on its streamline (𝑛 = constant) so that
𝑑𝑛
=0
𝑑𝑡
Chapter 3
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Hence,
𝑎=(
𝜕𝑣𝑠
𝜕𝑣𝑠
𝜕𝒔̂
𝜕𝒔̂
+ 𝑣𝑠
) 𝒔̂ + 𝑣𝑠 ( + 𝑣𝑠 )
𝜕𝑡
𝜕𝑠
𝜕𝑡
𝜕𝑠
Figure 4.9 Relationship between the unit vector along the streamline, 𝒔̂, and the radius of
curvature of the streamline, 𝕽
The quantity 𝜕𝒔̂⁄𝜕𝑠 represents the limit as 𝛿𝑠 → 0 of the change in the unit
vector along the streamline, 𝛿𝒔̂, per change in distance along the streamline, 𝛿𝑠.
The magnitude of 𝒔̂ is constant (|𝒔̂| = 1; it is a unit vector), but its direction is variable if the streamlines are curved.
𝛿𝒔̂ = 𝒔̂(𝑠 + 𝛿𝑠) − 𝒔̂(𝑠) = (𝒔̂ +
𝜕𝒔̂
𝜕𝒔̂
𝛿𝑠) − 𝒔̂ =
𝛿𝑠
𝜕𝑠
𝜕𝑠
In the limit 𝛿𝑠 → 0, the direction of 𝛿𝒔̂⁄𝛿𝑠 is seen to be normal to the streamline.
That is,
̂ = 𝛿𝜃𝒏
̂
𝛿𝒔̂ = |𝒔̂|𝛿𝜃𝒏
Thus, by equating the above two equations of 𝛿𝒔̂
𝜕𝒔̂
̂
𝛿𝑠 = 𝛿𝜃𝒏
𝜕𝑠
𝜕𝒔̂⁄𝜕𝑠 can be expressed as
Chapter 3
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
̂
𝜕𝒔̂ 𝛿𝜃
𝛿𝜃
𝒏
̂=
̂=
=
𝒏
𝒏
𝜕𝑠 𝛿𝑠
ℜ𝛿𝜃
ℜ
where 𝛿𝑠 = ℜ𝛿𝜃 from Fig. 4.9.
Similarly, the quantity 𝜕𝒔̂⁄𝜕𝑡 represents the limit as 𝛿𝑡 → 0 of the change
in the unit vector along the streamline, 𝛿𝒔̂, per change in time, 𝛿𝑡.
𝛿𝒔̂ = 𝒔̂(𝑡 + 𝛿𝑡) − 𝒔̂(𝑡) = (𝒔̂ +
𝜕𝒔̂
𝜕𝒔̂
𝛿𝑡) − (𝒔̂) =
𝛿𝑡
𝜕𝑡
𝜕𝑡
In the limit 𝛿𝑡 → 0, the direction of 𝛿𝒔̂⁄𝛿𝑡 is also seen to be normal to the
streamline. That is,
𝜕𝒔̂
̂
𝛿𝑡 = 𝛿𝜃𝒏
𝜕𝑡
Thus, as 𝛿𝑡 → 0
𝜕𝒔̂
𝛿𝜃
𝜕𝜃
̂=
̂
= lim
𝒏
𝒏
𝜕𝑡 𝛿𝑡→0 𝛿𝑡
𝜕𝑡
Finally by multiplying 𝑣𝑠 with 𝜕𝒔̂⁄𝜕𝑡 we get
𝑣𝑠
𝜕𝜃
𝜕𝑣𝑛
̂=
̂
𝒏
𝒏
𝜕𝑡
𝜕𝑡
̂ is directed from the streamline toward the center of
Note that the unit vector 𝒏
curvature.
Hence, the acceleration for two-dimensional flow can be written in terms of
its streamwise and normal components in the form
𝜕𝑣𝑠
𝜕𝑉
𝜕𝑣𝑛 𝑣𝑠2
̂
𝑎=(
+ 𝑣𝑠 ) 𝒔̂ + (
+ )𝒏
𝜕𝑡
𝜕𝑠
𝜕𝑡
ℜ
or
𝜕𝑣𝑠
𝜕𝑣𝑠
𝜕𝑣𝑛 𝑣𝑠2
𝑎𝑠 =
+ 𝑣𝑠
, 𝑎𝑛 =
+
𝜕𝑡
𝜕𝑠
𝜕𝑡
ℜ
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
6
The first term 𝜕𝑣𝑠 ⁄𝜕𝑡 represents the local acceleration in streamline direction, the second term 𝑣𝑠 𝜕𝑣𝑠 ⁄𝜕𝑠 represents the convective acceleration along the
streamline due to the spatial gradient i.e., convergence/divergence streamlines,
the third term 𝜕𝑣𝑛 ⁄𝜕𝑡 represents the local acceleration normal to flow, and the
last term 𝑣𝑠2 ⁄ℜ represents centrifugal acceleration (one type of convective acceleration) normal to the flow motion due to the streamline curvature.
3.3 Euler Equation
For an inviscid flow in which all the shearing stresses are zero (𝜇 = 0) the
general equations of motion (See Chapter 6) reduce to
𝜌𝑔 − ∇𝑝 = 𝜌𝑎
or
𝜌𝑔 − ∇𝑝 = 𝜌 [
𝜕V
+ (V ⋅ ∇)V]
𝜕𝑡
These equations are commonly referred to as Euler’s equations of motion, named
in honor of Leonhard Euler (1707 – 1783), a famous Swiss mathematician who pioneered work on the relationship between pressure and flow.
Although Euler’s equations are considerably simpler than the general equations of motion, they are still not amenable to a general analytical solution that
would allow us to determine the pressure and velocity at all points within an inviscid flow field. The main difficulty arises from the nonlinear velocity terms
(𝑢 𝜕𝑢⁄𝜕𝑥, 𝑣 𝜕𝑢⁄𝜕𝑥, etc.), which appear in the convective acceleration. Because
of these terms, Euler’s equations are nonlinear partial differential equations for
which we do not have a general method of solving. However, under some circumstances we can use them to obtain useful information about inviscid flow
fields. For example, as shown in the following section we can integrate the equations to obtain a relationship (the Bernoulli equation) between elevation, pressure, and velocity along a streamline.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
7
3.4 Bernoulli Equation
From Euler’s equation in vector form
𝜌𝑔 − ∇𝑝 = 𝜌 [
𝜕V
+ (V ⋅ ∇)V]
𝜕𝑡
we wish to integrate this differential equation along some arbitrary streamline
and select the coordinate system with the 𝑧 axis vertical (with “up” being positive)
so that the acceleration of gravity vector can be expressed as
𝑔 = −𝑔𝑘̂ = −𝑔∇𝑧
where 𝑔 is the acceleration of gravity vector. Also, it will be convenient to use the
vector identity
1
(V ⋅ ∇)V = ∇(V ⋅ V) − V × (∇ × V)
2
Euler equation can now be written in the form
𝜕V 1
−𝜌𝑔∇𝑧 − ∇𝑝 = 𝜌 [ + ∇(V ⋅ V) − V × (∇ × V)]
𝜕𝑡 2
and this equation can be rearranged to yield
𝜌
𝜕V
𝜌
+ ∇𝑝 + ∇(V 2 ) + 𝜌𝑔∇𝑧 = V × ω
𝜕𝑡
2
where 𝜔 = ∇ × V . If we restrict our attention to steady flow (𝜕⁄𝜕𝑡 = 0) and incompressible fluids (𝜌 = constant) Euler’s equation becomes
𝑝 1
∇ [ + V 2 + 𝑔𝑧] = V × ω
𝜌 2
or in a simple form,
∇𝐵 = V × ω
where 𝐵 ≡ 𝑝⁄𝜌 + 1⁄2 V 2 + 𝑔𝑧.
Chapter 3
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
1) Along a streamline
Along a streamline V × 𝒔̂ = 0 where 𝒔̂ is tangent to the streamline, and ∇𝐵
is perpendicular to V since V × ω is perpendicular to both V and ω. Thus,
𝒔̂ ⋅ ∇𝐵 =
𝜕𝐵
=0
𝜕𝑠
𝐵 is constant along the streamline. Thus, for inviscid, incompressible fluids
(commonly called ideal fluids) this can be written as
𝑝 𝑉2
+
+ 𝑔𝑧 = constant
𝜌 2
And this equation is the Bernoulli equation. It is often convenient to write the
equation between two points (1) and (2) along a streamline and to express the
equation in the “head” form by dividing each term by 𝑔 so that
𝑝1 𝑉12
𝑝2 𝑉22
+
+ 𝑧1 = +
+ 𝑧2
𝛾 2𝑔
𝛾 2𝑔
It should be emphasized that the Bernoulli equation is restricted to the following:




inviscid flow
steady flow
incompressible flow
flow along a streamline
2) Irrotational flow
For irrotational flow, 𝜔 ≡ ∇ × V = 0, so
∇𝐵 = 0
where 𝐵 = 𝑝⁄𝜌 + 1⁄2 V 2 + 𝑔𝑧 = constant and the constant is the same throughout the flow field. Thus for incompressible, irrotational flow the Bernoulli equation can be written as
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
9
𝑝1 𝑉12
𝑝2 𝑉22
+
+ 𝑧1 = +
+ 𝑧2
𝛾 2𝑔
𝛾 2𝑔
between any two points in the flow field. Above equation is restricted to




inviscid flow
steady flow
incompressible flow
irrotational flow
3) Unsteady irrotational flow
For an irrotational flow the velocity vector can be expressed in terms of a
scalar function 𝜙(𝑥, 𝑦, 𝑧, 𝑡) as
V = ∇𝜙
where 𝜙 is called the velocity potential (See Chapter 6). Then Euler’s equation
can be written as
∂ϕ 𝑝 1
∇ [ + + V 2 + 𝑔𝑧] = 0
∂t 𝜌 2
For inviscid, incompressible, irrotational, and unsteady fluids the Bernoulli
equation can be written as
𝑉2 𝑝
𝜙𝑡 +
+ + 𝑔𝑧 = F(𝑡)
2 𝜌
4) Streamline coordinates
As an alternate derivation, Euler’s equation for steady flow using streamline
coordinates becomes
𝜌𝑎 = −∇s (𝑝 + 𝛾𝑧)
where
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
10
𝜕𝑣𝑠
𝑣𝑠2
̂
𝑎 = 𝑣𝑠
𝒔̂ + 𝒏
𝜕𝑠
ℜ
∇s =
𝜕
𝜕
̂
𝒔̂ +
𝒏
𝜕𝑠
𝜕𝑛
Along a streamline
From the components of the Euler equation in streamline direction,
𝜌𝑣𝑠
𝜕𝑣𝑠
𝜕(𝑝 + 𝛾𝑧)
=−
𝜕𝑠
𝜕𝑠
This equation can be rearranged to yield
𝜕 𝑣𝑠2 𝑝
[ + + 𝑔𝑧] = 0
𝜕𝑠 2 𝜌
and can be integrated to give the Bernoulli equation along a streamline
𝑣𝑠2 𝑝
+ + 𝑔𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
2 𝜌
Normal to a streamline
From the components of the Euler equation normal to streamlines,
𝑣𝑠2
𝜕(𝑝 + 𝛾𝑧)
𝜌
=−
ℜ
𝜕𝑛
by rearranging and integrating the Bernoulli equation across streamlines is written as
𝑣𝑠2
𝑝
∫ 𝑑𝑛 + + 𝑔𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
ℜ
𝜌
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
11
3.5 Physical interpretation of Bernoulli equation
Integration of the equation of motion to give the Bernoulli equation actually corresponds to the work-energy principle often used in the study of dynamics.
This principle results from a general integration of the equations of motion for an
object in a very similar to that done for the fluid particle. With certain assumptions, a statement of the work-energy principle may be written as follows:
The work done on a particle by all forces acting on the particle is equal to
the change of the kinetic energy of the particle.
The Bernoulli equation is a mathematical statement of this principle.
In fact, an alternate method of deriving the Bernoulli equation is to use the
first and second laws of thermodynamics (the energy and entropy equations), rather than Newton’s second law. With the approach restrictions, the general energy equation reduces to the Bernoulli equation.
An alternate but equivalent form of the Bernoulli equation is
𝑝 𝑉2
+
+ 𝑧 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝛾 2𝑔
along a streamline.
Pressure head:
Velocity head:
𝑝
𝛾
𝑉2
2𝑔
Elevation head: 𝑧
The Bernoulli equation states that the sum of the pressure head, the velocity
head, and the elevation head is constant along a streamline.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
12
3.6 Static, Stagnation, Dynamic, and Total Pressure
1
𝑝 + 𝜌𝑉 2 + 𝛾𝑧 = 𝑝𝑇 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
2
along a streamline.
Static pressure: 𝑝
1
Dynamic pressure: 𝜌𝑉 2
2
Hydrostatic pressure: 𝛾𝑧
Stagnation points on bodies in flowing fluids.
1
Stagnation pressure: 𝑝 + 𝜌𝑉 2 (assuming elevation effects are negligible)
2
1
Total pressure: 𝑝𝑇 = 𝑝 + 𝜌𝑉 2 + 𝛾𝑧 (along a streamline)
2
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
The Pito-static tube (left) and typical Pitot-static tube designs (right).
Typical pressure distribution along a Pitot-static tube.
Chapter 3
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Chapter 3
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
3.7 Applications of Bernoulli Equation
1) Stagnation Tube
𝑉12
𝑉22
𝑝1 + 𝜌
= 𝑝2 + 𝜌
2
2
𝑉12 =
2
(𝑝 − 𝑝1 )
𝜌 2
=
2
(𝛾𝑙)
𝜌
𝑉1 = √2𝑔𝑙
𝑧1 = 𝑧2
𝑝1 = 𝛾𝑑, 𝑉2 = 0
𝑝2 = 𝛾(𝑙 + 𝑑) 𝑔𝑎𝑔𝑒
Limited by length of tube and need
for free surface reference
Chapter 3
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
2) Pitot Tube
𝑝1 𝑉12
𝑝2 𝑉22
+
+ 𝑧1 = +
+ 𝑧2
𝛾 2𝑔
𝛾 2𝑔
1
2
𝑉2 = {2𝑔 [(
𝑝1
𝑝2
+ 𝑧1 ) − ( + 𝑧2 )]}
𝛾
𝛾
⏟
⏟
ℎ1
ℎ2
where, 𝑉1 = 0 and ℎ = piezometric head
𝑉 = 𝑉2 = √2𝑔(ℎ1 − ℎ2 )
ℎ1 − ℎ2 from manometer or pressure gage
For gas flow Δ𝑝⁄𝛾 ≫ Δ𝑧
𝑉=√
2Δ𝑝
𝜌
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
16
3) Free Jets
Vertical flow from a tank
Application of Bernoulli equation between points (1) and (2) on the streamline
shown gives
1
1
𝑝1 + 𝜌𝑉12 + 𝛾𝑧1 = 𝑝2 + 𝜌𝑉22 + 𝛾𝑧2
2
2
Since 𝑧1 = ℎ, 𝑧2 = 0, 𝑉1 ≈ 0, 𝑝1 = 0, 𝑝2 = 0, we have
1
𝛾ℎ = 𝜌𝑉22
2
𝑉2 = √2
𝛾ℎ
= √2𝑔ℎ
𝜌
Bernoulli equation between points (1) and (5) gives
𝑉5 = √2𝑔(ℎ + 𝐻)
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
4) Simplified form of the continuity equation
Steady flow into and out of a tank
Obtained from the following intuitive arguments:
Volume flow rate: 𝑄 = 𝑉𝐴
Mass flow rate: 𝑚̇ = 𝜌𝑄 = 𝜌𝑉𝐴
Conservation of mass requires
𝜌1 𝑉1 𝐴1 = 𝜌2 𝑉2 𝐴2
For incompressible flow 𝜌1 = 𝜌2 , we have
𝑉1 𝐴1 = 𝑉2 𝐴2
or
𝑄1 = 𝑄2
Chapter 3
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Chapter 3
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
5) Volume Rate of Flow (flowrate, discharge)
1. Cross-sectional area oriented normal to velocity vector
(simple case where 𝑉 ⊥ 𝐴)
𝑈 = constant: 𝑄 = volume flux = 𝑈𝐴 [m/s  m2 = m3/s]
𝑈 ≠ constant: 𝑄 = ∫𝐴 𝑈𝑑𝐴
Similarly the mass flux = 𝑚̇ = ∫𝐴 𝜌𝑈𝑑𝐴
2. General case
𝑄 = ∫ V ⋅ 𝒏𝑑𝐴
𝐶𝑆
= ∫ |V| cos 𝜃 𝑑𝐴
𝐶𝑆
𝑚̇ = ∫ 𝜌(V ⋅ 𝒏)𝑑𝐴
𝐶𝑆
Average velocity:
𝑉̅ =
𝑄
𝐴
Chapter 3
19
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Example:
At low velocities the flow through a long circular tube, i.e. pipe, has a parabolic velocity distribution (actually paraboloid of revolution).
𝑟 2
𝑢 = 𝑢𝑚𝑎𝑥 (1 − ( ) )
𝑅
i.e., centerline velocity
a) find 𝑄 and 𝑉̅
𝑄 = ∫ V ⋅ 𝒏 𝑑𝐴 = ∫ 𝑢𝑑𝐴
𝐴
𝐴
2𝜋
∫ 𝑢𝑑𝐴 = ∫
𝐴
𝑅
∫ 𝑢(𝑟)𝑟𝑑𝜃𝑑𝑟
0
0
𝑅
= 2𝜋 ∫ 𝑢(𝑟)𝑟𝑑𝑟
0
2𝜋
where, 𝑑𝐴 = 2𝜋𝑟𝑑𝑟, 𝑢 = 𝑢(𝑟) and not 𝜃, ∴ ∫0 𝑑𝜃 = 2𝜋
𝑅
𝑟 2
1
𝑄 = 2𝜋 ∫ 𝑢𝑚𝑎𝑥 (1 − ( ) ) 𝑟𝑑𝑟 = 𝑢𝑚𝑎𝑥 𝜋𝑅2
𝑅
2
0
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
20
6) Flowrate measurement
Various flow meters are governed by the Bernoulli and continuity equations.
Typical devices for measuring flowrate in pipes.
Three commonly used types of flow meters are illustrated: the orifice meter, the nozzle meter, and the Venturi meter. The operation of each is based on
the same physical principles—an increase in velocity causes a decrease in pressure. The difference between them is a matter of cost, accuracy, and how closely
their actual operation obeys the idealized flow assumptions.
We assume the flow is horizontal (𝑧1 = 𝑧2 ), steady, inviscid, and incompressible between points (1) and (2). The Bernoulli equation becomes:
1
1
𝑝1 + 𝜌𝑉12 = 𝑝2 + 𝜌𝑉22
2
2
If we assume the velocity profiles are uniform at sections (1) and (2), the continuity equation can be written as:
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
21
𝑄 = 𝑉1 𝐴1 = 𝑉2 𝐴2
where 𝐴2 is the small (𝐴2 < 𝐴1 ) flow area at section (2). Combination of these
two equations results in the following theoretical flowrate
2(𝑝1 − 𝑝2 )
𝑄 = 𝐴2 √
𝜌[1 − (𝐴2 ⁄𝐴1 )2 ]
Other flow meters based on the Bernoulli equation are used to measure
flowrates in open channels such as flumes and irrigation ditches. Two of these
devices, the sluice gate and the sharp-crested weir, are discussed below under
the assumption of steady, inviscid, incompressible flow.
Sluice gate geometry
We apply the Bernoulli and continuity equations between points on the free surfaces at (1) and (2) to give:
1
1
𝑝1 + 𝜌𝑉12 + 𝛾𝑧1 = 𝑝2 + 𝜌𝑉22 + 𝛾𝑧2
2
2
and
𝑄 = 𝑉1 𝐴1 = 𝑏𝑉1 𝑧1 = 𝑉2 𝐴2 = 𝑏𝑉2 𝑧2
With the fact that 𝑝1 = 𝑝2 = 0:
Chapter 3
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
2𝑔(𝑧1 − 𝑧2 )
𝑄 = 𝑧2 𝑏√
1 − (𝑧2 ⁄𝑧1 )2
In the limit of 𝑧1 ≫ 𝑧2 :
𝑄 = 𝑧2 𝑏√2𝑔𝑧1
Rectangular, sharp-crested weir geometry
For such devices the flowrate of liquid over the top of the weir plate is dependent on the weir height, 𝑃𝑤 , the width of the channel, 𝑏, and the head, 𝐻, of
the water above the top of the weir. Between points (1) and (2) the pressure and
gravitational fields cause the fluid to accelerate from velocity 𝑉1 to velocity 𝑉2 . At
(1) the pressure is 𝑝1 = 𝛾ℎ, while at (2) the pressure is essentially atmospheric,
𝑝2 = 0. Across the curved streamlines directly above the top of the weir plate
(section a–a), the pressure changes from atmospheric on the top surface to some
maximum value within the fluid stream and then to atmospheric again at the bottom surface.
For now, we will take a very simple approach and assume that the weir flow
is similar in many respects to an orifice-type flow with a free streamline. In this
instance we would expect the average velocity across the top of the weir to be
proportional to √2𝑔𝐻 and the flow area for this rectangular weir to be proportional to 𝐻𝑏. Hence, it follows that
3
𝑄 = 𝐶1 𝐻𝑏√2𝑔𝐻 = 𝐶1 𝑏√2𝑔𝐻2
Chapter 3
23
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
3.8 Energy grade line (EGL) and hydraulic grade line (HGL)
In this chapter, we neglect losses and/or minor losses, and energy input or
output by pumps or turbines:
ℎ𝐿 = 0, ℎ𝑝 = 0, ℎ𝑡 = 0
𝑝1 𝑉12
𝑝2 𝑉22
+
+ 𝑧1 = +
+ 𝑧2
𝛾 2𝑔
𝛾 2𝑔
Define
𝑝
𝐻𝐺𝐿 = + 𝑧
𝛾
𝑝
𝑉2}
𝛾
2𝑔
𝐸𝐺𝐿 = + 𝑧 +
point-by point application is graphically displayed
HGL corresponds to pressure tap measurement + 𝑧
EGL corresponds to stagnation tube measurement + 𝑧
𝐸𝐺𝐿 = 𝐻𝐺𝐿 if 𝑉 = 0
ℎ𝐿 = 𝑓
𝐸𝐺𝐿1 = 𝐸𝐺𝐿2 + ℎ𝐿
𝐿 𝑉2
𝐷 2𝑔
i.e., linear variation in
𝐿 for 𝐷 , 𝑉 , and 𝑓
constant
for ℎ𝑝 = ℎ𝑡 = 0
𝑓 = friction factor
𝑓 = 𝑓(𝑅𝑒)
Pressure tap:
𝑝2
𝛾
=ℎ
Stagnation tube:
𝑝2
𝛾
+
𝑉22
2𝑔
=ℎ
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Helpful hints for drawing 𝐻𝐺𝐿 and 𝐸𝐺𝐿
1. 𝐸𝐺𝐿 = 𝐻𝐺𝐿 + 𝑉 2 ⁄2𝑔 = 𝐻𝐺𝐿 for 𝑉 = 0
2. 𝑝 = 0 ⇒ 𝐻𝐺𝐿 = 𝑧
3. for change in 𝐷 ⇒ change in 𝑉
i.e.
𝑉1 𝐴1 = 𝑉2 𝐴2
𝑉1
𝜋𝐷12
4
𝑉1 𝐷12
= 𝑉2
=
𝜋𝐷22
4
𝑉2 𝐷22
}⇒
Change in distance between
𝐻𝐺𝐿 & 𝐸𝐺𝐿 and slope change
due to change in ℎ𝐿
4. If 𝐻𝐺𝐿 < 𝑧 then 𝑝⁄𝛾 < 0 i.e., cavitation possible
Chapter 3
24
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
Chapter 3
25
Condition for cavitation:
𝑝 = 𝑝𝑣𝑎 = 2000 N⁄m2
Gage pressure 𝑝𝑣𝑎,𝑔 = 𝑝𝑣𝑎 − 𝑝𝑎𝑡𝑚 ≈ −𝑝𝑎𝑡𝑚 = −100,000 N⁄m2
𝑝𝑣𝑎,𝑔
≈ −10m
𝛾
where 𝛾 = 9810 N⁄m2
3.9 Limitations of Bernoulli Equation
Assumptions used in the derivation Bernoulli Equation:
(1) Inviscid
(2) Incompressible
(3) Steady
(4) Conservative body force
1) Compressibility Effects:
The Bernoulli equation can be modified for compressible flows. A simple,
although specialized, case of compressible flow occurs when the temperature of a
perfect gas remains constant along the streamline—isothermal flow. Thus, we
consider 𝑝 = 𝜌𝑅𝑇, where 𝑇 is constant (In general, 𝑝, 𝜌, and 𝑇 will vary). An
equation similar to the Bernoulli equation can be obtained for isentropic flow of a
perfect gas. For steady, inviscid, isothermal flow, Bernoulli equation becomes
𝑅𝑇 ∫
𝑑𝑝 1 2
+ 𝑉 + 𝑔𝑧 = 𝑐𝑜𝑛𝑠𝑡
𝑝 2
The constant of integration is easily evaluated if 𝑧1 , 𝑝1 , and 𝑉1 are known at some
location on the streamline. The result is
𝑉12
𝑅𝑇
𝑝1
𝑉22
+ 𝑧1 +
ln ( ) =
+ 𝑧2
2𝑔
𝑔
𝑝2
2𝑔
Chapter 3
26
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2007
2) Unsteady Effects:
The Bernoulli equation can be modified for unsteady flows. With the inclusion of the unsteady effect (𝜕𝑉 ⁄𝜕𝑡 ≠ 0) the following is obtained:
𝜌
𝜕𝑉
𝜕𝑡
1
𝑑𝑠 + 𝑑𝑝 + 𝜌𝑑(𝑉 2 ) + 𝛾𝑑𝑧 = 0 (along a streamline)
2
For incompressible flow this can be easily integrated between points (1) and (2) to
give
1
𝑠 𝜕𝑉
2
1
𝑝1 + 𝜌𝑉12 + 𝛾𝑧1 = 𝜌 ∫𝑠 2
𝜕𝑡
1
𝑑𝑠 + 𝑝2 + 𝜌𝑉22 + 𝛾𝑧2 (along a streamline)
2
3) Rotational Effects
Care must be used in applying the Bernoulli equation across streamlines. If
the flow is “irrotational” (i.e., the fluid particles do not “spin” as they move), it is
appropriate to use the Bernoulli equation across streamlines. However, if the
flow is “rotational” (fluid particles “spin”), use of the Bernoulli equation is restricted to flow along a streamline.
4) Other Restrictions
Another restriction on the Bernoulli equation is that the flow is inviscid. The
Bernoulli equation is actually a first integral of Newton's second law along a
streamline. This general integration was possible because, in the absence of viscous effects, the fluid system considered was a conservative system. The total energy of the system remains constant. If viscous effects are important the system
is nonconservative and energy losses occur. A more detailed analysis is needed
for these cases.
The Bernoulli equation is not valid for flows that involve pumps or turbines.
The final basic restriction on use of the Bernoulli equation is that there are no
mechanical devices (pumps or turbines) in the system between the two points
along the streamline for which the equation is applied. These devices represent
sources or sinks of energy. Since the Bernoulli equation is actually one form of
the energy equation, it must be altered to include pumps or turbines, if these are
present.