Determining surface pressure from skin friction

Abstract

This paper presents a general method to extract a surface pressure field from a skin friction field in complex flows as an inverse problem, focusing on its application to global luminescent oil-film (GLOF) skin friction measurements. The main technical aspects of this method are described, including the basic equation relating surface pressure to skin friction, variational method, numerical algorithm, and approximate method with the constant boundary enstrophy flux (BEF). The proposed method is evaluated through simulations in the Falkner–Skan flow and the flow over a 70°-delta wing to investigate the effects of the Lagrange multiplier, downsampling rate, noise level, and the value of a constant source term in the approximate method. Further, the approximate method is applied to the skin friction fields obtained by GLOF measurements in the flow over a 65°-delta wing and the square junction flow to obtain the normalized surface pressure fields. The proposed method provides a useful tool to obtain the high-resolution fields of both surface pressure and skin friction by GLOF measurements in complex flows (particularly at low speeds).

Graphical abstract

The normalized surface pressure field extracted from the skin friction field in the square junction flow, where skin friction lines are superposed.

Appendix: Neumann condition and Lagrange multiplier

The accuracy of the Neumann condition in the problem of surface pressure from skin friction is examined, and the effect of the Lagrange multiplier is discussed. As indicated in Sect. 2, the physically real surface quantities ${\varvec{\tau}}_{r}$, $p_{r}$ , and $\Phi_{r}$ satisfy the exact relation

$${\varvec{\tau}}_{r} \cdot {\mathbf{\nabla }}p_{r} = \Phi_{r} ,$$

(27)

where the subscript “r” denotes real quantities. In addition, from the NS equations, the pressure gradient on a surface (wall) is

$$\nabla \left[ {p_{r} } \right]_{w} = - {\varvec{n}} \times {\varvec{\sigma}} = \mu \left[ {\frac{{\partial^{2} {\varvec{u}}_{\pi } }}{{\partial n^{2} }}} \right]_{w} ,$$

(28)

where ${\varvec{\sigma}}$ is the boundary vorticity flux (BVF), ${\varvec{n}}$ is the unit normal vector of the surface, and ${\varvec{u}}_{\pi }$ is the velocity parallel to the surface. On the boundary curve (contour) $\partial D$ on the surface, from Eq. (27), we have

$$\frac{{\partial [p_{r} ]_{w} }}{{\partial \hat{n}}} = - \hat{\user2{n}} \cdot {\varvec{n}} \times {\varvec{\sigma}} = \hat{\user2{n}} \cdot \left[ {\mu \frac{{\partial ({\varvec{\omega}} \times {\varvec{n}})}}{\partial n}} \right]_{w} = \mu \left[ {\frac{{\partial^{2} ({\varvec{u}}_{\pi } \cdot \hat{\user2{n}})}}{{\partial n^{2} }}} \right]_{w},$$

(29)

where $\hat{\user2{n}}$ is the outward-pointing unit normal vector of $\partial D$. According to Eq. (29), the normal derivative of the physically real surface pressure at $\partial D$ (e.g., $\partial [p_{r} ]_{w} /\partial \hat{n}$) is determined by the BVF $\sigma$ or equivalently the boundary-normal diffusion of the surface-tangential velocity component ${\varvec{u}}_{\pi } \cdot \hat{\user2{n}}$.

In general, the right-hand side of Eq. (28) does not vanish automatically. Based on the fact that the BVF represents the boundary vorticity creation rate, the Neumann boundary condition $\partial \left[ {p_{r} } \right]_{w} /\partial \hat{n} \approx 0$ holds as an effective approximation for $\left\| {\varvec{\sigma}} \right\| \ll 1$. In other words, this approximation holds when the boundary vorticity creation rate at $\partial D$ does not play an important role for the region concerned. This could be more reasonable when the computational domain is sufficiently large. In image processing, the computational domain is usually much larger than the region of interest, and the side effect of the Neumann condition as an approximation on the result is small.

In the variational framework, the Euler–Lagrange equation for the approximate surface pressure $p$ is used as a model equation of Eq. (3), i.e.,

$${\varvec{\tau}} \cdot \nabla \left( {{\varvec{\tau}} \cdot \nabla p - \Phi } \right) + \left( {{\varvec{\tau}} \cdot \nabla p - \Phi } \right)\nabla \cdot {\varvec{\tau}} + \alpha {\kern 1pt} \nabla^{2} p = 0$$

(30)

with the Neumann boundary condition $\partial p/\partial \hat{n} = 0$ on $\partial D$. The approximate surface pressure is expressed as $p = p_{r} + \delta p$, where $p_{r}$ is the physically real surface pressure (the exact solution of the NS equations) and $\delta p$ is the error introduced by the modeling (approximation) in the variational framework. Similarly, the similar expression is $\Phi = \Phi_{r} + \delta \,\Phi$, where $\Phi_{r}$ and $\delta \,\Phi$ represent the physically real source term and the corresponding error, respectively. For simplicity, it is assumed that skin friction is accurate, i.e., ${\varvec{\tau}} = {\varvec{\tau}}_{r}$. In this case, when $\delta p \to 0$ and $\delta \,\Phi \to 0$, the Lagrange multiplier $\alpha$ should be sufficiently small such that the effect of the surface pressure Laplacian ${\mathbf{\nabla }}^{2} p$ can be minimized. Then, the limiting behavior of Eq. (30) is consistent with Eq. (27).

On the other hand, we consider the pressure Poisson equation

$${\mathbf{\nabla }}^{2} p_{r} + \frac{{\partial^{2} p_{r} }}{{\partial n^{2} }} = \rho \left( {\frac{1}{2}\omega^{2} - {\varvec{S}}:{\varvec{S}}} \right),$$

(31)

where ${\varvec{S}}$ is the strain rate tensor. Applying the pressure Poisson equation on the surface, we have

$${\mathbf{\nabla }}^{2} p_{r} = - \left[ {\frac{{\partial^{2} p_{r} }}{{\partial n^{2} }}} \right]_{w}$$

(32)

For most wall regions, the first-order wall-normal pressure derivative $\left[ {\partial p_{r} /\partial n} \right]_{w}$ is determined by the skin friction divergence ${\mathbf{\nabla }} \cdot {\varvec{\tau}}_{r}$ that is directly related to the skin friction topology. In contrast, the effect from $\left[ {\partial^{2} p_{r} /\partial n^{2} } \right]_{w}$ is relatively weak. For example, in unidirectional pressure-driven Poiseuille flow, this term is exactly zero. A simple analysis from Eq. (30) shows that the error $\delta p$ in most surface regions should satisfy a Poisson equation, i.e.,

$${\mathbf{\nabla }}^{2} \delta p \approx - \alpha^{ - 1} \left[ {{\varvec{\tau}}_{r} \cdot {\mathbf{\nabla }}({\varvec{\tau}}_{r} \cdot {\mathbf{\nabla }}\delta p - \delta \,\Phi ) + ({\varvec{\tau}}_{r} \cdot {\mathbf{\nabla }}\delta p - \delta \,\Phi ){\mathbf{\nabla }} \cdot {\varvec{\tau}}_{r} } \right],$$

(33)

Therefore, increasing the value of $\alpha$ tends to attenuate the source term in Eq. (33), thereby reducing the magnitude of the error $\delta p$ and leading to a smoother surface pressure distribution. Near the separation and attachment lines dominated by the skin friction divergence ${\mathbf{\nabla }} \cdot {\varvec{\tau}}_{r}$, Eq. (33) is reduced to

$${\mathbf{\nabla }}^{2} \delta p \approx \alpha^{ - 1} \delta \,\Phi {\mathbf{\nabla }} \cdot {\varvec{\tau}}_{r} .$$

(34)

For such structures, the reduction in the pressure error could be achieved by more physical modeling of the BEF and a suitable Lagrange multiplier. In summary, to meet the above requirements on the Lagrange multiplier, there is an optimal value of the Lagrange multiplier.