Compressible Navier Stokes in cylinder coordinates

• schettel
In summary: Thanks.In summary, Daniel was trying to help the person who contacted him, but they were not asking the right questions. He provided links to some resources that might help the person.
schettel
Hello,

I need the Navier Stokes equations for compressible flow (Newtonian fluid would be ok) in cylinder coordinates. Can anybody help?

Thanks

Ugh...I get queesy looking at that dex.

$$\rho (\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_r}{\partial \theta} - \frac{v^2_\theta}{r} + v_z \frac{\partial v_r}{\partial z}) = -\frac{\partial p}{\partial r} + \rho g_r + \mu [\frac{1}{r} \frac{\partial}{\partial r}(r \frac{\partial v_r}{\partial r}) - \frac{v_r}{r^2} +\frac{1}{r^2} \frac{\partial^2 v_r}{\partial \theta^2} - \frac{2}{r^2} \frac{\partial v_\theta}{\partial \theta} + \frac{\partial^2 v_r}{\partial z^2}]$$

Holly crud that's a lot of typing.

Angular (theta) Direction:
$$\rho (\frac{\partial v_\theta}{\partial t} + v_r \frac{\partial v_\theta}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_\theta}{\partial \theta} + \frac{v_r v_\theta}{r} + v_z \frac{\partial v_\theta}{\partial z}) = -\frac{1}{r} \frac{\partial p}{\partial \theta} + \rho g_\theta + \mu [\frac{1}{r} \frac{\partial}{\partial r}(r \frac{\partial v_\theta}{\partial r}) - \frac{v_\theta}{r^2} +\frac{1}{r^2} \frac{\partial^2 v_\thata}{\partial \theta^2} - \frac{2}{r^2} \frac{\partial v_r}{\partial \theta} + \frac{\partial^2 v_\theta}{\partial z^2}]$$

Z Direction:
$$\rho (\frac{\partial v_z}{\partial t} + v_r \frac{\partial v_z}{\partial r} + \frac{v_\theta}{r} \frac{\partial v_z}{\partial \theta} + v_z \frac{\partial v_z}{\partial z}) = -\frac{\partial p}{\partial z} + \rho g_z + \mu [\frac{1}{r} \frac{\partial}{\partial r}(r \frac{\partial v_z}{\partial r}) + \frac{1}{r^2} \frac{\partial^2 v_z}{\partial \theta^2} + \frac{\partial^2 v_z}{\partial z^2}]$$

That still doesn't help him too much.U assume the fluid to have an incompressible flow...

Daniel.

That's right, unfortunately. Thanks for the typing, anyway. And thanks for the links. I'll take it home on the weekend and try to figure it out myself. I'm bad at maths, though.

I'm sorry,but you haven't asked for some kindergarten stuff.You need to know what a gradient,curl,divergence,tensor,partial derivative,cylindric coordinate,... are.

I am urging you to read the construction of these equations in the 6-th volume of Landau & Lifschitz theoretical physics course:"Fluid Mechanics",Pergamon Press.Any edition.

Daniel.

You know...I didn't realize you had asked for compressible flow. My oops again. I really must learn how to read. Oh well. I had a nice exercise in LATex.

1. What is the Compressible Navier Stokes equation in cylinder coordinates?

The Compressible Navier Stokes equation in cylinder coordinates is a mathematical model that describes the behavior of fluid flow in a cylindrical system. It is a set of partial differential equations that take into account the effects of viscosity, pressure, and compressibility on the fluid's velocity and density.

2. Why is the Compressible Navier Stokes equation important in fluid dynamics?

The Compressible Navier Stokes equation is important in fluid dynamics because it allows scientists and engineers to accurately model and predict the behavior of fluids in real-world systems, such as aircraft engines, rocket propulsion, and weather patterns. It also provides a deeper understanding of the physical processes involved in fluid flow.

3. How does the Compressible Navier Stokes equation differ from the Incompressible Navier Stokes equation?

The main difference between the Compressible Navier Stokes equation and the Incompressible Navier Stokes equation is that the former takes into account the changes in density and pressure of a fluid, while the latter assumes the fluid is incompressible and therefore has a constant density. This makes the Compressible Navier Stokes equation more suitable for modeling high-speed and high-pressure flows.

4. What are some applications of the Compressible Navier Stokes equation in engineering?

The Compressible Navier Stokes equation has many applications in engineering, including the design and analysis of aircraft and spacecraft aerodynamics, the development of efficient and safe combustion systems, and the optimization of turbomachinery, such as turbines and compressors. It is also used in weather forecasting and climate modeling.

5. What are some challenges in solving the Compressible Navier Stokes equation?

Solving the Compressible Navier Stokes equation can be challenging due to its nonlinearity and the presence of complex boundary conditions. It also requires a high level of computational resources and numerical techniques to accurately simulate the behavior of fluids in real-world systems. Additionally, the equation itself is highly dependent on the initial and boundary conditions, making it difficult to generalize solutions for different scenarios.

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