samples_new/texts_merged/2909063.md

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y⁺ Calculation, Example 6D

Example 6D: Consider a high-velocity fluid over a flat plate. It is desired to find the thickness of the viscous sublayer at $y^+=1$. The fluid is H₂O at 395 K and 1 MPa. Its free stream velocity is 700 m/s, and has a boundary layer $\delta=0.1$ m.

Solutions:

1. Use the "Yplus_LIKE_Eddy_Scales_Book_Version.m" application found in my CFD/turbulence book, "Applied Computational Fluid Dynamics and Turbulence Modeling", Springer International Publishing, 1st Ed., ISBN 978-3-030-28690-3, 2019, DOI: 10.1007/978-3-030-28691-0.

or

2. Get a free copy of "Yplus_LIKE_Eddy_Scales_Book_Version.m" at www.cfdturbulence.com, or email me at tayloreddydk1@gmail.com.

or

3. Use the free $y^+$ estimation GUI tool offered by cfd-online, which is at http://www.cfd-online.com/Tools/yplus.php

or

4. Follow the step-by-step solution shown in the next slide. ---PAGE_BREAK---

$y^+$ Calculation, Example 6D

From $P$ and $T$, $\rho = 942 \text{ kg/m}^3$ and $\mu = 2.28 \times 10^{-4} \text{ kg/m-s}$.

$$v=\frac{\mu }{\rho }=\frac{2.28\times {10}^{-4}}{942}=2.43\times {10}^{-7}{\text{ m}}^2\mathrm{/}\text{s}v\; =\; \backslash frac\{\backslash mu\}\{\backslash rho\}\; =\; \backslash frac\{2.28\; \backslash times\; 10^\{-4\}\}\{942\}\; =\; 2.43\; \backslash times\; 10^\{-7\}\; \backslash text\{\; m\}^2/\backslash text\{s\}$$

$$C_f=[2{\log }_{10}(R{e}_x)-0.65{]}^{-2.3}=[2{\log }_{10}(2.87\times {10}^8)-0.65{]}^{-2.3}=1.60\times {10}^{-3}C\_f\; =\; [2\; \backslash log\_\{10\}(Re\_x)\; -\; 0.65]^\{-2.3\}\; =\; [2\; \backslash log\_\{10\}(2.87\; \backslash times\; 10^8)\; -\; 0.65]^\{-2.3\}\; =\; 1.60\; \backslash times\; 10^\{-3\}$$

$${\tau }_w=C_f\frac{\rho U_{\mathrm{\infty }}^2}{2}=1.60\times {10}^{-3}\frac{942\ast {700}^2}{2}=3.78\times {10}^5\backslash tau\_w\; =\; C\_f\; \backslash frac\{\backslash rho\; U\_\backslash infty^2\}\{2\}\; =\; 1.60\; \backslash times\; 10^\{-3\}\; \backslash frac\{942\; *\; 700^2\}\{2\}\; =\; 3.78\; \backslash times\; 10^5$$

$$u_{\ast }=\sqrt{\frac{{\tau }_w}{\rho }}=\sqrt{\frac{3.78\times {10}^5}{942}}=20.0u\_*\; =\; \backslash sqrt\{\backslash frac\{\backslash tau\_w\}\{\backslash rho\}\}\; =\; \backslash sqrt\{\backslash frac\{3.78\; \backslash times\; 10^5\}\{942\}\}\; =\; 20.0$$

$$y(\text{at }{y}^{+}=1)=\frac{y^{+}v}{u_{\ast }}=\frac{1\ast 2.43\times {10}^{-7}}{20}=1.22\times {10}^{-8}\text{ m}y(\backslash text\{at\; \}\; y^+=1)\; =\; \backslash frac\{y^+\; v\}\{u\_*\}\; =\; \backslash frac\{1\; *\; 2.43\; \backslash times\; 10^\{-7\}\}\{20\}\; =\; 1.22\; \backslash times\; 10^\{-8\}\; \backslash text\{\; m\}$$ ---PAGE_BREAK---

y⁺ Calculation, Example 6D Solutions

Approach 1 and 2 (the Matlab script, Yplus_LIKE_Eddy_Scales_Book_Version.m)

$$R{e}_x=2.89\times {10}^{8}Re\_x\; =\; 2.89\; \backslash times\; 10^8$$

$$y(\text{at }{y}^{+}=1)=1.23\times {10}^{-8}\text{ m}y(\backslash text\{at\; \}\; y^+=1)\; =\; 1.23\; \backslash times\; 10^\{-8\}\; \backslash text\{\; m\}$$

Approach 4 (previous slide)

$$R{e}_x=2.87\times {10}^{8}Re\_x\; =\; 2.87\; \backslash times\; 10^8$$

$$y(\text{at }{y}^{+}=1)=1.22\times {10}^{-8}\text{ m}y(\backslash text\{at\; \}\; y^+=1)\; =\; 1.22\; \backslash times\; 10^\{-8\}\; \backslash text\{\; m\}$$

Approach 3 (cfd-online tool)