Ideal means inviscid for you?? In such a case there is no BL.

But if you are addressing something different please detail you question.

A boundary layer always exists near the walls when viscous fluid prevails. Viscosity is the result of molecular interactions, not the geometric characteristics of the wall. The roughness of the wall additionally affects the boundary layer.

In classical theory of potential flows a stagnation point can be a source to create a boundary layer. So the formation of a boundary layer can depend on the geometry of the ambient laminar flow:

https://en.wikipedia.org/wiki/Stagnation_point_flow

Maybe magnus effect could be something you are interested in:

https://en.wikipedia.org/wiki/Magnus_effect

We all know this from swimming or trying to walk in a river. Ideally, there would be know force acting on our feet. Due to the formation of a boundary layer at stagnation points you can calculate, our feet feel an imbalance and the flow turns out to form a wake and be asymmetric.

@Filippo Maria Denaro yes i mean inviscid fluid . Does it mean that when an inviscid fluid moves over let say a stationary plate , no slip condition does not prevail ?

Thanks

Rochak Badyal

no matter about steady or unsteady assumption. The theory for the existence of a BL along a wall assumed a small but finite viscosity, that is a real fluid.

The ideal inviscid flow has a discontinuity on the tangential velocity at the wall.

Filippo Maria Denaro thanks a lot.

Dear Baydal

Since the ideal fluid has no viscosity the boundary layer formation will lose a crucial condition to form. So ideal fluid does not form a boundary layer.

Kind regards

Hello,

I feel the question is not complete. But If you assume an Ideal fluid , no BL is formed . In viscous flow when there is a fluid body interaction at the surface of the body you have BL for no slip condition at the wall.

This is a very good question. The ideal fluid is described by the Euler equations presented in the paper published by the Berlin Academy of Sciences in 1757. The original system doesn’t exhibit a turbulent behaviour and consequently there is no formation of the boundary layer. The actual fluids with non-vanishing viscosity are governed by the Navier-Stokes Equations (NSE) that are derived from the Euler equations by introducing the viscosity stress tensor (see for example Landau & Lifshitz Fluid Mechanics, Chapter 2, & 15). The resulting system exhibits a very rich class of turbulent flows.

The interesting situation occurs when the kinematic viscosity in the NSE is approaching zero. On the one hand we expect that the equation will converge to the Euler equations on the other hand we see that the Reynolds number is increasing drastically and that the solutions should be turbulent.

Obviously this is an antinomy.

The discussion of the possible solution of this problem is presented in the paper: “Existence of the passage to the limit of inviscid fluid” by D. S. Goldobin (the text is available on arXiv).

According to the above paper we can look at the boundary layer for the ideal fluid as the limit of the boundary layer of a viscous system with the kinematic viscosity approaching very small values. This methodology is still open to further research.

Boundary layer specifically is created by shear layers which is due to molecular interaction of a viscous flow. So ideal flow by definition is an inviscid flow, hence no boundary layer would exist in such flow.

ideal fluid have no viscosity so no boundary layer

Dear Rochak,

In a inviscid fluid, the boundary layer thickness vanishes. There is a slip boundary condition at the wall. The finite velocity at the wall implies that there is rotation in an infinitely thin "boundary layer". One can however take into account the influence of viscosity in such a model, by assuming that there is tangential flow separation at sharp edges. This can be argued by the fact that at such an edge the Laplacian of the velocity becomes infinitely large. Hence the vanishing viscosity mu times the Lalplacian \partial^2u/\partial x^2 can become finite and non-negligeable in the momentum equation. Assuming tangential flow separation implies that vorticity is injected into the flow as an infinitely thin shear layer: jump (discontinuity) in tangential velocity along a surface in the flow. The integral of the vorticity (the circulation) that is shed, corresponds to a change in the circulation of the vorticity, in the infinitely thin "boundary layer" on the body's surface. Consequently, in an ideal inviscid fluid model, one can by assuming a Kutta condition at sharp edges on the wall, take locally and globally the influence of viscosity into account. This "miracle" allows inviscid fluid theory to be useful in practice. This allows to predict the lift force on a wing (steady and unsteady). As the shedding of a shear layer also implies the shedding of energy in the vorticity mode, one can predict some aspects of flow resistance. This allows to predict flow resistance due to constrictions n a duct. Hence inviscid fluid theory can with a minor modification be a useful limit for very high Reynolds numbers flow. I have been using this to predict sound absorption or production in various flows. This model however is a "low Strouhal number" approximation (quasi-steady model for flow separation at sharp edges).

I hope this will allow you to understand that inviscid theory is meaningful.

Regards

Mico

A summary of my long answer would be; yes in an ideal (inviscid) fluid there are boundary layers but they are infinitely thin. Furthermore they influence becomes significant when they separate from the wall. In an inviscid fluid this can occur at sharp edges in the wall and this should be imposed in order to find a physically relevant approximation of high Reynolds number flows

Moreover, when carrying a numerical simulation of the flow in an inviscid fluid (Euler model), the "numerical viscosity" due to finite size of grid elements, will impose such a Kutta condition at sharp edges. One can force such a separation by adding to the actual geometry a small sharp edge to impose flow separation.

Your question is interesting

Avraham Hirschberg Thank you so much for your answer!

The most strange aspect of the model, is that in some cases it is very useful!

No because no-slip boundary condition does not exist due to absence of viscosity. Therefore there is no shearing stress in ideal fluid and no boundary layer.

Dear Rochak Badyal ,

The boundary layer appears when there is a no-slip condition at the interface between the fluid and its wall. You may assume the no-slip condition just for the real fluid, not for ideal fluid. As you know, when you assume the fluid is ideal, then it means that the fluid has no viscous effect or inviscid (non-viscous). In other words, the no-slip assumption is valid when you consider that the fluid has its viscosity effect. With this assumption then you may have the boundary-layer flow.

Thank you and have a nice Saturday night at home.

Best regards.

Nazar

Dear Mahabat and Nazaruddin,

I guess you did not understand that I was trying to explain you that when applying a Kutta condition at sharp edges, vorticity is shed and there is actually an infinitely thin boundary layer on the surface of a body in a ideal fluid with vanishing viscosity. The flow separation predicted by the Kutta condition is actually a viscous effect implemented in a theory with vanishing small viscosity. The prediction for the lift of an airfoil using this theory is quite accurate. Actually in numerical codes based on the frictionless fluid theory (Euler equations) the flow separation (Kutta condition) at sharp edges is induced by numerical errors due to the singularity of the flow (one call this numerical viscosity). The exact value of this viscosity is unimportant as long as flow separation occurs. The predicted lift for aircrafts is quite accurate.

Your approach to « ideal fluid » is dogmatic. You should focus on the question: does flow separation occurs ( either imposed theoretically or induced by numerical artifacts)? If flow separation does occur an ideal fluid model can capture a lot of the actual physics in the limit of high Reynolds numbers.

Regards

Mico

I think we should never forget that all about we are talking are "mathematical models" representing with different degree of approximation the reality.

Boundary layer is just a model, it represents a viscous layer in the classical Prandtl theory (do not forget that is a mathematical model that extends theoretically the BL up to infinity). In aerodynamics the ideal fluid assumption does not allow the viscosity to enter in and the production of vorticity is modelled in different ways, Kutta condition being one of them.

When we work with curved shock waves (for example at very high Mach number), vorticity is produced behind the shock even for inviscid fluid and that is somehow like a BL.

I do wonder if the boundary layer without the presence of viscosity would be meaningful. In fact, all mathematical models just try to imitate the rea-world phenomena in a simple form considering significant contributors only. To this end, inviscid fluid is being used where the tangential force is not dominant. However, for modeling of walls, several models have been proposed. plz, see guild dynamics by M. White. In my view, the answer is no.

We have to make use of potential flow or Navier stocked equation. In the first, basically the boundary layer needs to be emulated mathematically, while in the second, consideration of ideal fluid omits all components relevant to viscosity, and thus it is unlikely that boundary layer presents anywhere.