Abstract
LDA measurements are known as measurements of velocity components, as has been demonstrated by Eq. (3.39). Because these velocity components are found in the LDA coordinate system, it is always necessary to transform them into the flow field system.
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Notes
- 1.
Another possibility to transform the turbulent stresses from the non-orthogonal into the orthogonal coordinate system will be presented here based on Fig. 6.6. In applying Eq. (6.41) and (6.42) to velocity fluctuations in respective velocity components, one obtains
$$u^{\prime} = u^{\prime}_{\textrm{x}} \cos \varphi_{\textrm{u}} + u^{\prime}_{\textrm{y}} \sin \varphi_{\textrm{u}}$$$$v^{\prime} = u^{\prime}_{\textrm{x}} \cos \varphi_{\textrm{v}} + u^{\prime}_{\textrm{y}} \sin \varphi_{\textrm{v}}$$Consequently there is
$$u^{\prime}v^{\prime} = u^{\prime\,2}_{\textrm{x}} \cos \varphi_{\textrm{u}}\cos \varphi_{\textrm{v}} + u^{\prime\,2}_{\textrm{y}} \sin \varphi_{\textrm{u}} \sin \varphi_{\textrm{v}} + u^{\prime}_{\textrm{x}}u^{\prime}_{\textrm{y}} \sin \left( {\varphi_{\textrm{u}} + \varphi_{\textrm{v}}} \right)$$The sample means of corresponding fluctuations are calculated as
$$\begin{aligned}&\sigma_{{\textrm{uu}}} = \overline {u^{\prime 2}} = \sigma_{\textrm{xx}} \cos ^2 \varphi_{\textrm{u}} + \sigma_{{\textrm{yy}}} \sin^2 \varphi_{\textrm{u}} + \tau_{{\textrm{xy}}} \sin 2\varphi_{\textrm{u}} \\ &\sigma_{{\textrm{vv}}} = \overline{v^{\prime 2}} = \sigma_{{\textrm{xx}}} \cos ^2 \varphi_{\textrm{v}} + \sigma_{{\textrm{yy}}} \sin^2 \varphi_{\textrm{v}} + \tau_{{\textrm{xy}}} \sin 2\varphi_{\textrm{v}} \\ &\tau_{{\textrm{uv}}} = \sigma_{{\textrm{xx}}} \cos \varphi_{\textrm{u}} \cos \varphi_{\textrm{v}} + \sigma_{{\textrm{yy}}} \sin \varphi_{\textrm{u}} \sin \varphi_{\textrm{v}} + \tau_{{\textrm{xy}}} \sin \left( {\varphi_{\textrm{u}} + \varphi_{\textrm{v}}} \right)\end{aligned}$$From these three equations with \(\sigma_{{\textrm{uu}}}\), \(\sigma_{{\textrm{vv}}}\) and \(\tau_{{\textrm{uv}}}\) as given quantities, the turbulent stresses \(\sigma_{{\textrm{xx}}}\), \(\sigma_{{\textrm{yy}}}\) and \(\tau_{{\textrm{xy}}}\) can be resolved. The same results as given in Eqs. (6.48), (6.49), and (6.50) are obtained.
References
Tropea C (1983) A note concerning the use of a one-component LDA to measure shear stress terms. Technical Notes, J Exp Fluids 1:209–210
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Zhang, Z. (2010). Linear Transformation of Velocities and Turbulent Stresses. In: LDA Application Methods. Experimental Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13514-9_6
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DOI: https://doi.org/10.1007/978-3-642-13514-9_6
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