This document discusses the impact of liquid jets on flat and curved plates that may be stationary or moving. It covers the following key points: 1. Equations are derived for the force exerted on a stationary flat plate based on momentum and Bernoulli's theorem. 2. The concept of a moving control volume is introduced to analyze the force on a moving flat plate from a jet. 3. Velocity triangles are used to determine the absolute velocity of the jet as it leaves a curved plate that may be stationary or moving.

1. Impact of Jet
COURSE COORDINATOR - DR. V.R. KALAMKAR ,
DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
• Flat Plate, Curved Plate
• Stationary and Moving Plate

2. Liquid Jet
COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
A stream of liquid flowing through some kind of orifice, nozzle,
aperture when projected to surrounding medium. The liquid
coming out is in the form of a Jet.
Jet[1]
1. Anderson Jr, J.D., 2010. Fundamentals of aerodynamics. Tata McGraw-Hill Education.

3. Impact of Jets
COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
If a plate is placed in its path, it will exert force on the plate or vane .
Depending upon the flat properties, different cases are possible like
1. Stationary Plate
2. Inclined or Vertical
3. Flat or Curved Plate

4. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
Force exerted on stationary flat plate(inclined)
V is the jet velocity
Q is Volume Flow Rate
D is the diameter of the jet
Θ is the inclination angle
A it the area of jet
Other symbols have standard notation.
Assumptions:-
1. Fluid is inviscid
2. Pressure is constant
3. Elevation difference between inlet and outlet point is
neglected
By Bernoulli Theorem,
Inlet Velocity = Outflow Velocity

5. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
• In order to calculate the amount of force on plate, the
reference axes are taken along the plate surface (OS)
and perpendicular(ON) to plate surface.
Along the OS plane, only frictional force is acting and
they are negligible.
Thus, Net force parallel to plate i.e. OS is Zero
Fs = 0
Momentum Theorem in OS direction,
𝜌𝑄2 𝑉 + 𝜌𝑄1 −𝑉 − 𝜌𝑄𝑉𝑐𝑜𝑠𝜃 = 𝐹𝑠 =0
𝑄2 − 𝑄1 = 𝑄 𝑐𝑜𝑠𝜃
As 𝑄2 + 𝑄1 = 𝑄 ;
Thus, 𝑄2 =
𝑄
2
1 + 𝑐𝑜𝑠𝜃 𝑜𝑟 𝑄1 =
𝑄
2
(1 + 𝑐𝑜𝑠𝜃)

6. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
Forces in a perpendicular plane ON
𝜌𝑄2 0 + 𝜌𝑄1 0 − 𝜌𝑄𝑉𝑠𝑖𝑛𝜃 = 𝐹𝑛;
𝐹𝑛 = −𝜌𝑄𝑉𝑠𝑖𝑛𝜃
Negative sign indicates the opposite direction.
Thus force acting on the plane;
𝐹𝑝= 𝐹𝑠 + 𝐹𝑛;
𝑭 𝒑 = 𝝆𝑸𝑽𝒔𝒊𝒏𝜽
If stationary plate is allowed to move with u
velocity, then work done rate will be:
𝜕𝑊
𝜕𝑡
= 𝑃 = 𝐹𝑃 𝑠𝑖𝑛𝜃. 𝑢 = ρ𝑄𝑉|𝑢|𝑠𝑖𝑛2
𝜃
u
This equation cannot be written for already moving
plate as force exerted on plate will be different.

7. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
Moving Flat plate
Consider a flat plate moving with velocity U as shown in figure.
θ
V is the jet velocity
Q is Volume Flow Rate
D is the diameter of the jet
Θ is the inclination angle
A it the area of jet
Other symbols have standard notation.

8. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
Moving Flat plate
This plate can be observed from stationary or moving
reference.
Consider a fixed Point O’ , when the moving plate is
observed from this point the flow structure at O’ keeps
changing and thus flow is unsteadiness w.r.t O’.
But if we observe the plate from a point the plate or moving
volume , this concept is called moving control volume.

9. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
To calculate the force exerted by jet on moving
plate resolve the forces along 2 axes;
Moving Flat plate
Consider a coordinate system in moving plane and
corresponding axes are ON or OS as shown in figure.
𝐴𝑙𝑜𝑛𝑔 𝑂𝑆, 𝐹𝑠 = 0;
𝐴𝑙𝑜𝑛𝑔 𝑂𝑁, 𝐹 𝑁 = 𝜌𝑄 𝑉 − 𝑢 𝑠𝑖𝑛𝜃
𝑄 = 𝑉 − 𝑢 𝑎
If V= u; Q = 0 ( No flow)
If V < u; Q < 0 ( No flow)
𝐹 𝑁 = 𝜌 𝑉 − 𝑢 𝑎 𝑉 − 𝑈 𝑠𝑖𝑛𝜃
or
Thus , 𝑭 𝑵 = 𝝆𝒂 𝑽 − 𝑼 𝟐
𝒔𝒊𝒏𝜽
Power, 𝑷 = 𝝆𝒂 𝑽 − 𝒖 𝟐 𝒖 𝒔𝒊𝒏𝜽

10. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
Velocity Triangle at Outlet
(𝑉 − 𝑢) 𝑖𝑠 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑉𝑅.
𝑉𝑅 is velocity with which fluid
comes out after striking the
plate.
For shockless, flow, 𝑉𝑅 should
be parallel to solid surface of
the plate.
The fluid comes out with a
absolute velocity 𝑉𝑜 its
magnitude and direction are
given by velocity triangle.

11. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
Impact of jet on Curved plate (moving)
Consider a curved plate moving with velocity u, jet
comes in with velocity V1 and leaves with V2 .
A control volume is considered which is moving with
velocity u in shown direction, the exerted force is
calculated in direction of moving velocity i.e. OX
direction. Force in OY direction is negligible.
𝑉1 is inlet velocity.
𝑉2 is outlet velocity.
U is blade velocity.
∝1 𝑖𝑠 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑖𝑛𝑙𝑒𝑡 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑉1 𝑎𝑛𝑑 𝑢 𝑤ℎ𝑒𝑟𝑒𝑎𝑠 ∝2 𝑖𝑠 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑛 𝑜𝑢𝑡𝑙𝑒𝑡 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑛𝑑 𝑏𝑙𝑎𝑑𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑢.
𝛽1 𝑎𝑛𝑑 𝛽2 𝑎𝑟𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑎𝑛𝑑 𝑏𝑙𝑎𝑑𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦.

12. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
𝐹𝑥 = 𝜌𝑄 −𝑉𝑟2 𝑐𝑜𝑠𝛽2 − 𝜌𝑄(𝑉𝑟1 𝑐𝑜𝑠𝛽1) ;
𝐹𝑥 = −𝜌𝑄(𝑉𝑟1 𝑐𝑜𝑠𝛽1 + 𝑉𝑟2 𝑐𝑜𝑠𝛽2)
Thus force acting on plate is given by
𝑭 𝑷𝒙= 𝝆𝑸(𝑽 𝒓𝟏 𝒄𝒐𝒔𝜷 𝟏 + 𝑽 𝒓𝟐 𝒄𝒐𝒔𝜷 𝟐).
Impact of jet on Curved plate (moving)
Force exerted in x- direction is given by

13. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
Inlet Velocity Triangle
Tangential component of 𝑽 𝟏 = 𝑽 𝒓𝟏 𝒄𝒐𝒔𝜷 𝟏 = 𝑽 𝒘𝟏
Outlet Velocity Triangle
Tangential component of 𝑽 𝟐 = 𝑽 𝒓𝟐 𝒄𝒐𝒔𝜷 𝟐 = 𝑽 𝒘𝟐
Thus, force equation will become;
𝑭 𝑷𝒙= 𝝆𝑸(𝑽 𝒘𝟏 + 𝑽 𝒘𝟐)

14. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
1. Select the correct assumptions for analysing flow of a jet that is deflected by a fixed or moving vane:
a. The moment of the jet is fixed.
b. The absolute speed does not change along the vane.
c. The fluid flows on to the vane without shock.
d. Friction between jet and vane is neglected
e. The velocity is uniform over the cross-section of the jet before and after contacting the vane.
Problems

15. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
2. When a steady inviscid jet impinges on a fixed inclined plane surface.
a. The momentum in the direction of the approach velocity is unchanged.
b. Force component along the plane surface exerted on the jet by the surface is zero.
c. The flow is divided into parts directly proportional to the angle of inclination of the surface.
d. The speed is reduced for that portion of the jet turned through more than 90° and increased for the other
portion.
e. The momentum component is unchanged parallel to the surface.

16. COURSE COORDINATOR - DR. V.R. KALAMKAR, DEPARTMENT OF MECHANICAL ENGINEERING,
VNIT, NAGPUR
3. A jet of water issuing from a nozzle strikes a flat plate normally .The plate moves towards the nozzle at a velocity of
10m/s while the jet issuing from the nozzle has a velocity of 15m/s. Compute the absolute velocity components of
the jet (in m/s) as it leaves the vane, parallel(x component) and normal (y component) to the undistributed jet.
Neglect friction on the surface.
a. 𝑣 𝑥 = 5, 𝑣 𝑦 = 5;
b. 𝑣 𝑥 = −5, 𝑣 𝑦 = 5;
c. 𝑣 𝑥 = 10, 𝑣 𝑦 = 10;
d. 𝑣 𝑥 = −10, 𝑣 𝑦 = 25;
e. 𝑣 𝑥 = 15, 𝑣 𝑦 = 15;