In this A 2D Navier-stokes equation is used to study steady state laminar flow over a backward facing step for dimension less duct. The Reynolds number used for this flow is Re=Uh/ν where Reh=229 for Denham and Patrick and Reh=178 and 233 for Aung’s Experiment. i.e. The results obtained are then compared using the excel software for which the data of Velocity profile and Reattachment length for the temperature profile in both recirculating and non-recirculating zone by using Para view. (Salome 8.3.0, sim flow)

1. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
LAMINAR FLOW OVER A BACKWARD-FACING STEP IN A
TWO- DIMENSIONAL FLOW CHANNEL
Introduction:
In this A 2D Navier-stokes equation is used to study steady state laminar flow over a
backward facing step for dimension less duct. The Reynolds number used for this flow is Re=Uh/ν
where Reh=229 for Denham and Patrick and Reh=178 and 233 for Aung’s Experiment. i.e. The
results obtained are then compared using the excel software for which the data of Velocity profile
and Reattachment length for the temperature profile in both recirculating and non-recirculating
zone by using Para view. (Salome 8.3.0, sim flow)
Reynolds Number=U*h/v
Where:
U= maximum inlet velocity
h=height of the inlet channel
v = Kinematic viscosity
Backward Facing Step Flow:
This takes place in two recirculating zones; the first zone occurs after the step from the
inlet and the second zone occurs on the upper wall after the recirculation of the first zone. The
point to know is the formation of profiles for the dependence on Reynolds Number.
Formulation:
We know that from the assumptions of the fluid
the strain rate is= > dß/dt = V/l
since velocity profile is in steady condition the velocity gradient will be du/dy=v/l
The fluid considered is Newtonian, and the shear is directly proportional to the strain rate for such
fluids i.e. τ = μ*du/dy
μ = Viscosity (Pa. s)

2. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
The equation for fluid in motion from the system of reference Eulerian which is also defined from
Gauss theorem is
ⅆ𝐺
ⅆ𝑡
=
𝜕𝜌𝑔
𝜕𝑡
+ div(𝜌𝑔𝑢̅)
The mass conservation equation:
𝐷𝑀
𝐷𝑡
= 0
𝐷𝐺
𝐷𝑡
=
𝜕𝜌𝑔
𝜕𝑡
+ div(𝜌𝑔𝑢̅)
𝜕𝜌
𝜕𝑡
+ div(𝜌𝑢̅) = 0
𝜌 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝜕𝜌
𝜕𝑡
= 0 ; div(𝑢̅) =
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
+
𝜕𝑤
𝜕𝑧
= 0
From Momentum Equation [ div(u)=0]
F=ma
𝐹̅ = 𝑚
ⅆ𝑢̅
ⅆ𝑡
=
𝐷( 𝑚𝑢̅)
𝐷𝑡
𝐷(𝑚𝑢̅)
𝐷𝑡
=
𝐷(𝜌𝑣𝑢̅)
𝐷𝑡
= 𝑣
𝐷(𝜌𝑢̅)
𝐷𝑡
𝑓𝑟𝑜𝑚 =>
𝐷𝐺
𝐷𝑡
=
𝜕𝜌𝑔
𝜕𝑡
+ div(𝜌𝑔𝑢̅)
G = m.u
g = u

3. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
i.e. the Force vector equations in X, Y, Z directions are: the forces are per unit volume.
𝐹𝑥
𝑣
=
𝜕𝜌𝑢
𝜕𝑡
+ div(𝜌𝑢𝑢̅)
𝐹𝑦
𝑣
=
𝜕𝜌𝑣
𝜕𝑡
+ div(𝜌𝑣𝑢̅)
𝐹𝑧
𝑣
=
𝜕𝜌𝑤
𝜕𝑡
+ div(𝜌𝑤𝑢̅)
the fluid flowing through the pipe, if we consider a small differential fluid particle the vector
equations will be:
𝐹𝑥 = −
𝜕𝜌
𝜕𝑥
+
𝜕𝜎𝑥𝑥
𝜕𝑥
+
𝜕 𝝉 𝑦𝑥
𝜕𝑦
+
𝜕 𝝉 𝑧𝑥
𝜕𝑧
𝐹𝑥 = −
𝜕𝜌
𝜕𝑦
+
𝜕𝜎𝑦𝑦
𝜕𝑦
+
𝜕 𝝉 𝑥𝑦
𝜕𝑥
+
𝜕 𝝉 𝑧𝑦
𝜕𝑧
𝐹𝑧 = −
𝜕𝜌
𝜕𝑧
+
𝜕𝜎𝑧𝑧
𝜕𝑧
+
𝜕 𝝉 𝑥𝑧
𝜕𝑥
+
𝜕 𝝉 𝑦𝑧
𝜕𝑦
This Equations are due to the viscosity of the fluid.
From Stoke’s law for the Newtonian fluids the following linear stress and shear stresses will be
occurred, and the following deformation equations are
𝜎𝑥𝑥 = 2𝜇
𝜕𝑢
𝜕𝑥
𝜎𝑦𝑦 = 2𝜇
𝜕𝑣
𝜕𝑦
𝜎𝑧𝑧 = 2𝜇
𝜕𝑤
𝜕𝑧
𝝉 𝑥𝑦 = 𝝉 𝑦𝑥 = 𝜇
𝜕𝑢
𝜕𝑦
+
𝜕𝑣
𝜕𝑥
𝝉 𝑥𝑦 = 𝝉 𝑦𝑥 = 𝜇
𝜕𝑢
𝜕𝑦
+
𝜕𝑣
𝜕𝑥

4. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
𝜎𝑥𝑥 = 2𝜇
𝜕𝑢
𝜕𝑥
+ 𝜆 div(𝑢̅)
The linear stress for which applied stress is material constant. i.e. linear deformation and the
viscous deformation and the constant λ will be the second viscosity i.e. 2/3μ
We know that as the Reynolds number increase with the lesser the thickness.
Fig.1. 3-Dimensional Naiver Stokes Equations

5. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
Boundary Conditions:
Fig-2: -Setup parameters for the pipe
Fig-3: -Mesh Generation:
Fig-4: - Mesh with uniform Grid
The mesh is selected in the form to see the refined grid with number of nodes but the mesh
which is refined can be with lower deviation error so for the refines mesh the nodes here is
55450.
To check the mesh is refined mesh sensitivity analysis is taken in to picture and the
deviation error is calculated for 4 different nodes mesh and the deviation error is less than 1% i.e.
0.9 % for 55450 nodes mesh this analysis is considered in both the re circulating zones.
As the increase in the number of nodes make the mesh more refined.
When we consider mesh 1&2
% 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 =
𝐌𝐞𝐬𝐡 𝟐 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲 − 𝐦𝐞𝐬𝐡 𝟏 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲
𝐑𝐞𝐟𝐞𝐫𝐞𝐧𝐜𝐞 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲

6. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
In our case the reference velocity will be 1 because of the dimensionless condition.
MESH1-15544 Nodes
MESH2-30982 Nodes
Mesh3-45252 Nodes
Mesh4-55450 Nodes {MAXIMUM and Refined Mesh}
Fig 5: - velocity of fluid in para view software

7. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
The inlet condition for the velocity of fluid is =
-0.9318*y^4 + 7.6704*y^3 - 24.112*y^2 + 33.969*y - 16.588
Fig 8: - Reference point showing with velocity profile in para view
The flow of fluid velocity form inlet to outlet with some points of reference to extract the
velocity profiles and temperature profiles data in the form of excel.csv file and by using the axis
of values we generate the profiles.
Fig 6: - Boundary condition for adiabatic walls Fig 7: - Boundary condition for inlet velocity

8. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
Results:
Velocity Profiles:
Mesh Sensitivity Analysis:

9. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
Reattachment Lengths:
(for Reynolds Number 233)
(for Reynolds Number 178)
Temperature Profiles:
For Re=178

10. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
For Re=233
Conclusion:
The solution obtained was in good agreement by comparing the reference of Denham and Patrick
for velocity profiles and Aung’s experiment for temperature profile at Reynolds number
178&233. By the above results the obtained profiles are for temperature is also good at re-
attachment lengths.

11. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
References:
Reynolds number = 178 comparison of Temperature profiles for Backward facing step with
Aung’s experiment.

12. NUMERICAL HEAT AND MASS TRANSFER
BY: -MD. ABDUL
UNIVERSITY OF CASSINO AND SOUTHERN LAZIO
Reynolds number = 233 comparison of Temperature profiles for Backward facing step with
Aung’s experiment.