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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-13514-9_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13513-2
Online ISBN: 978-3-642-13514-9
eBook Packages: EngineeringEngineering (R0)