General method for determining the boundary layer thickness in nonequilibrium flows

While the computation of the boundary-layer thickness is straightforward for canonical equilibrium flows, there are no established definitions for general nonequilibrium flows. In this paper, a method is developed based on a local reconstruction of the inviscid velocity profile $U_I\left[y\right]$ resulting from the application of the Bernoulli equation in the wall-normal direction. The boundary-layer thickness ${\delta }_{99}$ is then defined as the location where $U/U_I=0.99$, which is consistent with its classical definition for the zero-pressure-gradient boundary layers. The proposed local-reconstruction method is parameter-free and can be deployed for both internal and external flows without resorting to an iterative procedure, numerical integration, or numerical differentiation. The superior performance of the local-reconstruction method over various existing methods is demonstrated by applying the methods to laminar and turbulent boundary layers and two flows over airfoils. Numerical experiments reveal that the local-reconstruction method is more accurate and more robust than existing methods, and it is applicable for flows over a wide range of Reynolds numbers.

Figure 1

Included are a profile from the suction side of a NACA 4412 airfoil [22] at the angle of attack (AoA) $=5^{\circ }$ and the Reynolds number based on the chord length ${\text{Re}}_c={10}^6$ at the streamwise location $x/c=0.72$ (upper) and a profile from a NACA 0012 airfoil [23] at AoA $=0^{\circ }$ and ${\text{Re}}_c=400000$ at the streamwise location $x/c=0.90$ (lower). Distributions of the mean streamwise velocity (black solid lines) from simulations and the generalized velocity (blue dashed lines) computed using Eq. (2) versus the wall-normal coordinate $y$ normalized by the airfoil chord $c$. Blue triangles indicate the locations $y={\delta }_{99}$ identified from Eq. (3). Note that $U_p$ denotes the velocity at the furthest point from the wall in the wall-normal profile.

Figure 2

Distributions of the mean $z$ vorticity ${\mathrm{\Omega }}_z$ (black solid lines) and the boundary layer sensor $-y{\mathrm{\Omega }}_z$ [10] (red dashed lines) versus the wall-normal coordinate $y$ normalized by $c$. The red diamonds indicate the locations $y={\delta }_{99}$ identified from Eq. (4). Included are a profile from the suction side of a NACA 4412 airfoil at AoA $=5^{\circ }$ and ${\text{Re}}_c={10}^6$ [22] at the streamwise location $x/c=0.72$ (left curves and symbols) and a profile from a NACA 0012 airfoil at AoA $=0^{\circ }$ and ${\text{Re}}_c=400000$ [23] at the streamwise location $x/c=0.90$ (right curves and symbols).

Figure 3

Distributions of the stagnation pressure $P_o$ and the dynamic pressure (a), and the velocity profile $U$ and the locally reconstructed inviscid velocity profile $U_I$ (b) versus the normalized wall-normal coordinate $y/c$. The maximum stagnation pressure (a) and the boundary-layer edge (b) are indicated with the black hexagram and the green square, respectively. The data is from the suction side of a NACA 4412 airfoil at AoA $=5^{\circ }$ and ${\text{Re}}_c={10}^6$ [22] at the streamwise station $x/c=0.20$.

Figure 4

Distributions of the stagnation pressure $P_o$ and the dynamic pressure (a), and the velocity profile $U$ and the locally reconstructed inviscid velocity profile $U_I$ (b) versus the normalized wall-normal coordinate $y/c$. This is the same as Fig. 3 except that one profile is from the suction side of a NACA 4412 airfoil at AoA $=5^{\circ }$ and ${\text{Re}}_c={10}^6$ [22] at the streamwise station $x/c=0.72$ (left curves and symbols) and the other profile is from a NACA 0012 airfoil at AoA $=0^{\circ }$ and ${\text{Re}}_c=400000$ [23] at the streamwise station $x/c=0.90$ (right curves and symbols). The maximum stagnation pressure (a) and the boundary-layer edge (b) are indicated with the black hexagram and the green square, respectively.

Figure 5

Distributions of the relative error between the computed and the classical definitions of the boundary-layer thickness versus the Reynolds number ${\text{Re}}_x$ for the Blasius boundary layer. Included are the results from the ${\tilde{U}}_{\infty }$ method (blue dotted line), the $-y{\mathrm{\Omega }}_z$ threshold method (red dash-dotted line), and the local-reconstruction method (green solid line) for computing $\delta$ and a $1/{\text{Re}}_x$ reference (black dashed line). Note that the integral in the ${\tilde{U}}_{\infty }$ method is artificially truncated at the analytical boundary-layer edge so a converged solution is possible for this flow.

Figure 6

Distributions of the relative error between the computed and the classical definitions of the boundary-layer thickness versus the friction Reynolds number ${\text{Re}}_{\tau }$. Included are the results from the local-reconstruction method (green squares), the ${\tilde{U}}_{\infty }$ method (blue triangles), the $-y{\mathrm{\Omega }}_z$ threshold method (red diamonds), and the mean-shear threshold method (purple circles). The cases plotted are ZPGBLs of Refs. [17, 28, 30]. The mean-vorticity-based methods cannot be deployed with the data of Ref. [30] because the vorticity statistics are not reported.

Figure 7

Distributions of the mean streamwise velocity (black solid lines) versus the wall-normal coordinate $y$ for various profiles from the suction side of a NACA 4412 at AoA $=5^{\circ }$ and ${\text{Re}}_c={10}^6$ detailed in Table 1. These wall-normal profiles originate from the airfoil surface at the streamwise stations $x/c=0.20,0.37,0.55,0.72,0.90$ (from top left to bottom right). The estimates of the boundary-layer edges (and the corresponding edge velocities) are plotted with symbols as indicated in the legend.

Figure 8

Distributions of the mean streamwise velocity (black solid lines) versus the wall-normal coordinate $y$ for various profiles from a symmetric NACA 0012 at AoA $=0^{\circ }$ and ${\text{Re}}_c=400000$ detailed in Table 1. These wall-normal profiles originate from the airfoil surface at the streamwise stations $x/c=0.21,0.37,0.55,0.72,0.90$ (from top left to bottom right). The estimates of the boundary-layer edges (and the corresponding edge velocities) are plotted with symbols as indicated in the legend.

Figure 9

Distributions of the mean streamwise velocity (black solid lines) versus the wall-normal coordinate $y$ for various profiles from of a NACA 4412 at AoA $=5^{\circ }$ and ${\text{Re}}_c={10}^6$. This is the same as Fig. 7 except that included profiles are from the pressure side of the airfoil (see Table 1 for details) and the wall-normal profiles originate from the airfoil surface at the streamwise stations $x/c=0.21,0.37,0.54,0.72,0.89$ (from left to right). The estimates of the boundary-layer edges (and the corresponding edge velocities) are plotted with symbols as indicated in the legend.

Figure 10

Distributions of the mean streamwise velocity $U$ and the generalized velocity $\tilde{U}$ (a), the mean vorticity ${\mathrm{\Omega }}_z$, the $-y{\mathrm{\Omega }}_z$ boundary layer sensor, and the threshold constant (b), the stagnation pressure $P_o$ and the dynamic pressure $0.5\rho U_m^2$ (c), and the mean velocity $U$ and the locally reconstructed inviscid solution $U_I$ (d) versus the wall-normal coordinate $y$ for the pressure side of a NACA 4412 airfoil at AoA $=5^{\circ }$ and ${\text{Re}}_c={10}^6$ detailed in Table 1. All data is from the streamwise location $x/c=0.21$. The estimates of the boundary-layer edges from the respective methods (a), (b), (d) and maximum stagnation pressure (c) are plotted with symbols. The subscript $p$ denotes quantities evaluated at the furthest point from the wall in the wall-normal profile.

Figure 11

This is the same as Fig. 10 except that the data is taken from wall-normal profiles originating from the airfoil surface at the streamwise station $x/c=0.89$. The estimates of the boundary-layer edges from the respective methods (a), (b), (d) and maximum stagnation pressure (c) are plotted with symbols.

Figure 12

Distributions of the mean streamwise velocity (solid lines) in panel (a), and the stagnation (solid lines) and dynamic pressure (dashed lines) in panel (b) versus the wall-normal coordinate $y$ for various streamwise stations in a turbulent APGBL [6] with details specified in Table 1. These wall-normal profiles originate from the airfoil surface at the streamwise stations $x/L_x=0.09,0.27,0.44,0.62,0.79,0.97$ (from left to right), where $L_x$ denotes the length of the plate. The subscript $L_y$ refers to quantities evaluated on the domain boundary at $y=L_y$. In panel (a), the estimates of the boundary-layer edges (and the corresponding edge velocities) are plotted with symbols as indicated in the legend. In panel (b), the maxima of the stagnation pressure profiles are also indicated with symbols.

Figure 13

Distributions of the mean streamwise velocity $U$ and the generalized velocity $\tilde{U}$ (a), the mean vorticity ${\mathrm{\Omega }}_z$, the $-y{\mathrm{\Omega }}_z$ boundary layer sensor, and the threshold constant (b), the mean velocity $U$ and the locally reconstructed inviscid solution $U_I$ (d) versus the wall-normal coordinate $y$ for the turbulent APGBL [6] detailed in Table 1. Panel (b) is reproduced in panel (c) on zoomed linear axes. All data is taken from the streamwise station at $x/L_x=0.27$. The estimates of the boundary-layer edges (and the corresponding edge velocities) from the respective methods are plotted with symbols.

Figure 14

Distributions of the velocity profile $U$, the locally reconstructed inviscid velocity profile $U_I$, and the hyperbolic and linear approximations of $U_I$ versus the normalized wall-normal coordinate $y/c$. The boundary-layer thicknesses implied by Eq. (9) based on the three approximate inviscid solutions are indicated with squares, diamonds, and circles, respectively. All data are from a NACA 4412 airfoil at AoA $=5^{\circ }$ and ${\text{Re}}_c={10}^6$ [22]. Specifically, the data are extracted from the suction side at the streamwise station $x/c=0.72$ (a) and the pressure side at $x/c=0.21$ (b).