Hi there,
In a turbulent motion the length scale (l) and the time scale (t) of the smallest eddy motions vary according to:
length scale: l/L = Re^(-0.75)
time scale t/(L/U) = Re^(-0.5)
(actually there should b that symbol for proportionality rather than the equal sign in the above two expressions...dont know how to use that symbol here!)
where the Reynold's no Re=(L*speed/kinematic viscosity)
Question: How do the a) number of grid nodes and b) the number of time steps required to compute the properties of a turbulent flow depend on the Reynold's number?
I tried substituing 1000 for the Re number in the above two expressions for length and time scales but am not sure wat to do after tht. I guess if someone knowldegable cud explain how the node or time scales are worked out n used for computation ill better understand the question n hopefully then work the answer out. I just think I don't know much abt how the nodes n time step things work.
Thanks,
AKHTAR
To understand how the number of grid nodes and time steps required to compute the properties of a turbulent flow depend on the Reynolds number, let's break it down.
a) Number of grid nodes:
In computational fluid dynamics (CFD), the domain is discretized into a grid of cells or nodes. The number of grid nodes influences the resolution and accuracy of the simulation. Generally, as the Reynolds number increases, the number of grid nodes needed also increases.
To explain the relationship between the Reynolds number (Re) and the number of grid nodes, we need to consider the length scale (l) mentioned in your question. The length scale is related to the size of the smallest eddies in the turbulent flow.
According to the given expression: l/L = Re^(-0.75), where l is the length scale and L is the characteristic length of the flow, we can rearrange it to get L = l/(Re^(-0.75)).
Now, let's assume that we want a certain level of resolution for our simulation, meaning that we want to resolve the smallest eddies up to a certain length scale. Let's call this desired length scale l_desired.
To ensure that the simulation has enough grid nodes to capture these eddies, we need to choose the size of the domain (L) accordingly. From the equation above, we can see that as the Reynolds number increases (Re), the characteristic length (L) decreases. This implies that the number of grid nodes required increases to maintain the desired resolution.
In summary, as the Reynolds number increases, the number of grid nodes required to compute the properties of a turbulent flow typically increases to maintain the desired level of resolution.
b) Number of time steps:
Simulating turbulent flows also requires dividing time into discrete steps. The number of time steps needed depends on the time scale (t) mentioned in your question.
The equation given for the time scale is: t/(L/U) = Re^(-0.5), where t is the time scale, L is the characteristic length, and U is the characteristic velocity of the flow.
Similar to the relationship between the Reynolds number and the number of grid nodes, we can see that as the Reynolds number increases, the characteristic length (L) decreases. This implies that the number of time steps required to simulate the flow also increases to accurately capture the dynamics of turbulence.
The time step size is determined by the Courant-Friedrichs-Lewy (CFL) condition, which ensures numerical stability in the simulation. In general, as the Reynolds number increases, the CFL condition requires smaller time steps, thus increasing the number of time steps needed.
In conclusion, as the Reynolds number increases, both the number of grid nodes and the number of time steps required to compute the properties of a turbulent flow typically increase to maintain the desired resolution and stability in the simulation.