Courses by Software
Courses by Semester
Courses by Domain
Tool-focused Courses
Machine learning
POPULAR COURSES
Post Graduate Program in Hybrid Electric Vehicle Design and AnalysisPost Graduate Program in Computational Fluid DynamicsPost Graduate Program in CADPost Graduate Program in CAEPost Graduate Program in Manufacturing DesignPost Graduate Program in Computational Design and Pre-processingPost Graduate Program in Complete Passenger Car Design & Product DevelopmentSuccess Stories
RANS
Let us consider the Navier Stokes equation for a 2D, incompressible flow. The continuity equation for this can be written as below. Applying Reynold's decomposition by substituting , Eqn(1) becomes …
Abdul Rehman Sadiq K
updated on 15 Feb 2020
Let us consider the Navier Stokes equation for a 2D, incompressible flow. The continuity equation for this can be written as below.
∂u∂x+∂v∂y=0→Eqn(1)∂u∂x+∂v∂y=0→Eqn(1)
Applying Reynold's decomposition by substituting u=(ˉu+u′)andv=(ˉv+v′)u=(¯u+u')andv=(¯v+v'), Eqn(1) becomes
∂ˉu∂x+∂u′∂x+∂ˉv∂y+∂v′∂y=0→Eqn(2)∂¯u∂x+∂u'∂x+∂¯v∂y+∂v'∂y=0→Eqn(2)
Integrating with respect to time 't' from 0 to t,
1t∫t0∂ˉu∂xdt+1t∫t0∂u′∂xdt+1t∫t0∂ˉv∂ydt+1t∫t0∂v′∂ydt=0→Eqn(3)1t∫t0∂¯u∂xdt+1t∫t0∂u'∂xdt+1t∫t0∂¯v∂ydt+1t∫t0∂v'∂ydt=0→Eqn(3)
Since time integral of fluctuating component is zero, ∫t0∂u′∂xdt=0and∫t0∂v′∂ydt=0∫t0∂u'∂xdt=0and∫t0∂v'∂ydt=0
∴Eqn(3)⇒1t∂ˉu∂x∫t0dt+1t∂ˉv∂y∫t0dt=0→Eqn(4)
∂ˉu∂x(t-0t)+∂ˉv∂y(t-0t)=0→Eqn(5)
∂ˉu∂x+∂ˉv∂y=0→Eqn(6)
Eqn(6) is the Reynold's Averaged continuity equation.
The momentum equation in the x- direction can be written as follows.
∂u∂t+u∂u∂x+v∂u∂y=-1ρ∂P∂x+ν∂2u∂y2→Eqn(7)
In order to simplify, let us add u(∂u∂x+∂v∂y) to the LHS. From Eqn(1) this term is zero
∂u∂t+u∂u∂x+v∂u∂y+u(∂u∂x+∂v∂y)⏟0=-1ρ∂P∂x+ν∂2u∂y2→Eqn(8)
∂u∂t+2u∂u∂x+v∂u∂y+u∂v∂y=-1ρ∂P∂x+ν∂2u∂y2→Eqn(9)
Consider ∂(uv)∂y=u∂v∂y+v∂u∂yand∂(u2)∂x=2u∂u∂x, substituting these in Eqn(9)
∂u∂t+∂(u2)∂x+∂(uv)∂y=-1ρ∂P∂x+ν∂2u∂y2→Eqn(10)
Applying reynold's decomposition and integrating with respect to time from 0 to t,
1t∫t0∂ˉu∂tdt+1t∫t0∂u′∂tdt+1t∫t0∂(ˉu)2∂xdt+1t∫t0∂(u′)2∂xdt+1t∫t0∂(2ˉuu′)∂xdt+1t∫t0∂(ˉuˉv)∂ydt+1t∫t0∂(ˉuv′)∂ydt+1t∫t0∂(u′ˉv)∂ydt+1t∫t0∂(u′v′)∂ydt=-1ρ(1t∫t0∂ˉP∂x+1t∫t0∂P′∂x)+ν(1t∫t0∂2ˉu∂y2+1t∫t0∂2u′∂y2)→Eqn(11)
Rearranging the terms
∂ˉu∂t+∂(ˉu)2∂x+∂(ˉuˉv)∂y=-1ρ∂ˉP∂x+ν∂2ˉu∂y2-1t∫t0∂(u′)2∂xdt-1t∫t0∂(u′v′)∂ydt→Eqn(12)
It can be proven by order of magnitude analysis that ∫t0∂(u′)2∂xdt is negligibly small, and can be equated to zero.
∴∂ˉu∂t+∂(ˉu)2∂x+∂(ˉuˉv)∂y⏟Inertial Terms=-1ρ∂ˉP∂x⏟Pressure Term+1ρ∂∂y(μ∂ˉu∂y⏟Viscous Stress-ρt∫t0(u′v′)dt⏟Reynold's Stress)⏟Diffusion Term→Eqn(13)
Eqn(13)is the Reynold's Averaged momentum equation in the X direction. Applying Reynold's decomposition for Y direction would result in similar equation.
Comparing Eqn(10) and Eqn(13), it can be observed that there are some extra terms which are dependent on the fluctuating components of velocity. The time integral ρt∫t0∂(u′v′)∂ydt is called as Reynold's Stress. This term gives the value of momentum diffusivity due to turbulence. Reynold's stress is the term which simplifies the Navier-Stokes equation and represents the For a 3D flow, there would be another term, ρt∫t0∂(u′w′)∂zdt. It can be observed that in the RANS momentum equation, the derivative of fluctuating components only in the perpedicaular directions .
Leave a comment
Thanks for choosing to leave a comment. Please keep in mind that all the comments are moderated as per our comment policy, and your email will not be published for privacy reasons. Please leave a personal & meaningful conversation.
Other comments...
Be the first to add a comment
Read more Projects by Abdul Rehman Sadiq K (20)
a new title
Interpolation schemes in FVM:The flux terms in FVM are calculated using the values at the cell centres. For example, heat flux is calculated using temperature values at cell centres (E,P and W). But the flux is obtained when we multiply this by thermal diffusivity. Thermal diffusivity depends on temperature and the value…
15 Feb 2020 02:26 AM IST
RANS
Let us consider the Navier Stokes equation for a 2D, incompressible flow. The continuity equation for this can be written as below. Applying Reynold's decomposition by substituting , Eqn(1) becomes …
15 Feb 2020 02:26 AM IST
Prandtl-Meyer Expansion Shock problem using Converge
In this project the Prandtl-Meyer expansion fan was simulated to study compressible flow. Air flowing with a velocity of 680m/s (M=2) was simulated over a wedge to study the expansion. Shocks are created when an object moves at a speed greater than that of sound in that medium. Shock are sudden change in the physical properties…
15 Feb 2020 02:26 AM IST
Effects of Air preheating on Combustion
Introduction
15 Feb 2020 02:26 AM IST
Related Courses
0 Hours of Content
Skill-Lync offers industry relevant advanced engineering courses for engineering students by partnering with industry experts.
Our Company
4th Floor, BLOCK-B, Velachery - Tambaram Main Rd, Ram Nagar South, Madipakkam, Chennai, Tamil Nadu 600042.
Top Individual Courses
Top PG Programs
Skill-Lync Plus
Trending Blogs
© 2025 Skill-Lync Inc. All Rights Reserved.