Computational and Theoretical Analyses of the Precessing Vortex Rope in a Simplified Draft tube of a Scaled Model of a Francis Turbine


Significance Statement

Recent studies on draft tube surge have shown that Francis-type hydroturbines operating at off-design conditions are laden with numerous undesirable effects in the form of violent pressure variations caused by helical vortices generated in the draft tube. The helical vortex, called a vortex rope, revolves around the axis of the draft tube and hence is called a “precessing” fluid structure with high vorticity. At off-design conditions, hydroturbines are characterized by high swirl levels at the inlet of the draft tube. Also, a shear layer forms between a centrally stalled region typical of diminished axial velocities and the surrounding vicinity with substantially high axial flow velocities. A combination of these two effects may at times yield a region of pressure lower than the vapor pressure of water thereby leading to cavitation.

The vortex rope manifests itself by means of high amplitude and low frequency pressure fluctuations that are nearly periodic in nature. These fluctuations are extremely intense and induce severe vibrations and noise in the draft tube thereby disrupting the functioning of the turbine and reducing its efficiency, which calls for increased maintenance.

Pennsylvania State University researchers Dr. Girish Rajan and Professor John Cimbala analyzed the vortex rope extensively to gain information that may be used in its mitigation and minimization of its harmful effects on the turbine. They were also interested in knowing the conditions under which a vortex rope would be classified as weak or strong and estimated the critical state at which a transition would be expected. Their work is now published in Journal of Fluids Engineering.

The researchers started by investigating swirling flows in the simplified draft tube of a scaled model Francis turbine operating at constant head with its runner rotating at constant speed. They then considered four discharge coefficients for the numerical analysis. Later on, they used a three-dimensional, unsteady, Navier–Stokes solver with the detached-eddy simulation model and the realizable k–є model to analyze the vortex rope formed at different discharge coefficients. The dominant amplitude of the pressure fluctuations at a fixed points on the draft tube periphery and the size of the vortex rope were obtained for the individual cases.

The authors of this paper observed that the dominant amplitude of the pressure fluctuations at a fixed point in the draft tube increased by 13 times and the length of the rope increased by 3.4 times when the operating point of the turbine shifted from a discharge coefficient of 0.37 to 0.34. The analysis enabled them to more easily comprehend how strong a vortex rope gets as the operating point of the turbine shifts from its best efficiency point. It was also noted that pressure was not a suitable parameter for visualizing the vortex rope. The other chosen parameters (described in the paper) depicted the rope more prominently, and the structures of the vortex rope that were obtained using  these parameters were nearly identical to one another when comparing both large-scale and small-scale features.

The perturbation analysis presented in this research includes the formulation of a Sturm-Liouville (SL) system, and the identification of a critical discharge coefficient,  above which the solutions to the SL system are oscillatory and non-oscillatory otherwise. The results of this work are  promising  and have the potential to help in deciding the range within which the turbine may be operated, so that the undesirable effects of the vortex rope are minimal.

Computational and Theoretical Analyses of the Precessing Vortex Rope in a Simplified Draft tube of a Scaled Model of a Francis Turbine - advances in enginering Rajan_Cimbala_Vortex_Rope_figure_ALL

Precessing Vortex Rope in a Simplified Draft tube of a Scaled Model of a Francis Turbine - advances in engineering

About the author

Girish Kumar Rajan is an Assistant Professor of Mechanical Engineering at the Indian Institute of Petroleum and Energy (IIPE), Andhra Pradesh. In 2004, he joined the National Institute of Technology at Tiruchirappalli, from where he received his B. Tech. degree in Mechanical Engineering four years later. He was awarded the ‘Summer Undergraduate Research Grant for Excellence (SURGE)’ fellowship by the Indian Institute of Technology at Kanpur in 2007. As a SURGE fellow, he investigated the convective heat transfer due to fluid flow past discrete heater sources placed in a channel using computational fluid dynamics (CFD).  In 2008, Rajan completed his senior thesis, which contains the results of a numerical investigation of fluid flow in a Y-shaped divided intake duct.

Rajan moved to the United States in Fall 2008 to join the Pennsylvania State University (Penn State), and was honored with the Dean’s fellowship for the academic year 2008-2009 by the College of Engineering.  He received his M.A. degree in Mathematics (2012) and his M.S. degree in Mechanical Engineering (2013) from Penn State. In 2014, Rajan was honored with a Graduate Teaching Fellowship, which provided him with an opportunity to assume full responsibility for teaching a course in thermal-fluid sciences.

He graduated from Penn State with a Ph.D. in Mechanical Engineering in Spring 2015. Between 2008 and 2015, Dr. Rajan completed three major research projects at Penn State – an investigation of surface gravity waves propagating on water of finite, variable depth; an investigation of the vortex rope formed in a simplified draft tube of a scaled model of a Francis turbine using CFD; and the theoretical and experimental modeling of the dissipation rates of capillary-gravity waves.

In 2015, Dr. Rajan joined the William G. Pritchard Fluid Mechanics Laboratory housed in the Department of Mathematics at Penn State as a postdoctoral research scholar, and conducted theoretical and experimental research on waves at a contaminated interface separating two arbitrary fluids. After a nineteen-month stint as a postdoctoral fellow, Dr. Rajan moved to India and accepted an offer from IIPE. At IIPE, Dr. Rajan teaches courses in fluid mechanics, and conducts research in basic fluid mechanics, including interfacial flows and vortex flows.

Results of Dr. Rajan’s research have been presented in international conferences such as the annual meetings of the American Physical Society (APS) – Division of Fluid Dynamics (DFD); and published in reputed journals such as the Physics of Fluids, the Journal of Fluids Engineering and the SIAM Journal on Applied Mathematics (SIAP).

About the author

John M. Cimbala is Professor of Mechanical Engineering at The Pennsylvania State University (Penn State). He received his B.S. degree in Aerospace Engineering (1979) from Penn State. Then he obtained his M.S. degree (1980) and his Ph.D. degree (1984) in Aeronautics from The California Institute of Technology (Caltech). In 1984, Dr. Cimbala returned to Penn State as Assistant Professor of Mechanical Engineering. In 1990, he was promoted to Associate Professor and was granted tenure. In 1997, he was promoted to Professor.

During a sabbatical leave in 1993-94, Professor Cimbala worked at NASA Langley Research Center in Hampton, VA, where he advanced his knowledge of computational fluid dynamics (CFD) and turbulence modeling. During a sabbatical leave in 2002-03, he co-authored an undergraduate textbook, “Fluid Mechanics: Fundamentals and Applications”, Y. A. Çengel and J. M. Cimbala, McGraw-Hill, New York (2006), now in its fourth edition (2018); it is used throughout the world, and has been translated into several languages. Professor Cimbala is author or co-author of several other textbooks and dozens of journal and conference papers. While on sabbatical leave during the academic year 2010-11, he worked at American Hydro Corporation in York, PA, where he used CFD to model large hydroturbines.

Dr. Cimbala conducts experimental and computational research in basic fluid mechanics, turbulence, and turbomachinery. He teaches courses in fluid mechanics; indoor air quality; instrumentation, measurements, and statistics; and air pollution. He has been an educational innovator throughout his career, such as using and promoting others to use the Internet, tablet PCs, and cell phone feedback in undergraduate and graduate courses. Awards include: College of Engineering Outstanding Teaching Award (1992), College of Engineering Premier Teaching Award (1996), George W. Atherton Award for Excellence in Teaching (1997), Teacher of the Year Award from Pi Tau Sigma (1997), and College of Engineering Outstanding Advising Award (1998). In 2009 he became a Fellow of the American Society of Mechanical Engineers.

On the personal side, Dr. Cimbala is married with two adult sons and one grandson, and enjoys working out, swimming, pool (billiards), and family activities. He is an avid reader, and he especially enjoys studying, memorizing, and teaching the Bible. In June 2015 he published his first novel through Kindle Direct Publishing, “I, Adam: The Man without a Navel.”



Girish K. Rajan, John M. Cimbala. Computational and Theoretical Analyses of the Precessing Vortex Rope in a Simplified Draft tube of a Scaled Model of a Francis Turbine. Journal of Fluids Engineering February 2017, Volume 139.

Department of Mechanical Engineering, Pennsylvania State University, University Park, PA 16802.

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