تحلیل دو و سه بعدی جریان‏ فوق بحرانی در خم‏ها

جریان فوق بحرانی در کانال‏های افقی با خم 90 درجه‏ و مقطع مستطیلی به صورت آزمایشگاهی و عددی بررسی می‌شود. نسبت شعاع انحناء به عرض کانال (rc/b) در محدوده 1.50 تا 4.83 و عدد فرود جریان ورودی به خم fr0 بین 1.82 تا 6.18 قرار دارد. در شبیه‏سازی دو بعدی از مدل عددی roe2d و برای مدل‏سازی سه بعدی از نرم‏افزار flow3d استفاده شده است. مدل‏ سه بعدی به خوبی قادر به شبیه‏سازی رفتار جریان فوق بحرانی در خم است اما مدل دو بعدی که از حل معادلات آب‏های کم‏عمق به دست آمده است، برای مقادیر 3 > fr0 دقت قابل قبولی دارد. با افزایش عدد فرود جریان ورودی به خم، خطای این مدل بیشتر می‏شود. با مقایسه توزیع فشارهای هیدرواستاتیک و هیدرودینامیک و همچنین مولفه قائم شتاب در خم ثابت می‌شود که فرض اساسی معادلات آب‏های کم‏عمق یعنی توزیع فشار هیدرواستاتیک که در نتیجه آن از اثر شتاب قائم ذرات آب صرفنظر می‏شود، تاثیر زیادی در خطای برآورد رفتارجریان به خصوص در اعداد فرود بالا دارد.

Two- and Three-Dimensional Analysis of Supercritical Flow in Bends

Introduction: Channel bends are sometimes unavoidable due to project conditions or land topography. However, oblique cross waves are a distinct feature of supercritical flow in bends. These waves continue for a long distance downstream and increase the height of water considerably. Initially, the complex behavior of supercritical flow in bends was studied by hydraulic models in the laboratory. Later on, numerical models were found inexpensive tools to investigate flow patterns and explain features that may not even be possible to measure. In this article, the supercritical flow in a rectangular horizontal channel of 90º bend is studied with different ratios of radius to channel width (rc/b) using two and threedimensional numerical models. Water surface profiles are then compared with the data that were obtained from our experimental bend models. It is proved that threedimensional models are more successful in predicting the flow profile, peak, and location of waves at the outer wall bend.Methodology: In this study, Flow3D was used for the threedimensional simulation of flow patterns. This software had a wide variety of applications and capabilities. The user could enter information to select different models to provide a range of flow phenomena. Flow3D integrated the NavierStokes equations (NS) with finite volume method (FVM), with different mesh configurations, suitable for complex geometries. The kε turbulence model was used to close the NS partial differential equations. The volume of fluid (VOF) method was used to model the free surface boundary. Additional boundary conditions for supercritical flow in bends included constant depth and velocity at the inflow section and noslip or zero velocity conditions at the floor and solid walls. The Roe2D model was used for the simulation of twodimensional shallow water equations. This model was able to capture discontinuities such as shock waves in supercritical flow. A triangular mesh was used for the space discretization, and a minmod slope limiter was implemented to control oscillations. Experiments were performed in the curved channel of the hydraulic laboratory of Ferdowsi University of Mashhad. This rectangular channel was horizontal, 40 cm in width, and the walls and floor were made of transparent plexiglass sheets. A straight channel, 1.8 m length, was installed before the bend to ensure flow development length. At the end of this channel was the 90º channel bend with internal and external radii of 40 and 80 cm, respectively. The channel width could be changed by adding interior walls; thereby, the ratio of rc/b might be changed accordingly. Results and Discussion: Several experiments were run in the curved channels with widths of 15, 20, 30, and 40 cm and different radius of curvature to channel width (rc/b). The flow rate and water depth were measured, and thereby, the approach Froude number Fro was calculated. New experimental equations were obtained to calculate the maximum flow depth and location of the first wave’s crest along the outer wall in terms of the approach Froud number and the geometric specification of the bend. For each experiment, the corresponding two and threedimensional computer models were performed too. The threedimensional model was well able to estimate the behavior of the supercritical flow, including the depth and position of wave crest at the outer wall of the bend. As Fro increased or rc/b decreased, the wave peak increased and moved downstream. However, the twodimensional model had acceptable accuracy only for low values of Fr0 < 3. the assumption of hydrostatic pressure in depthaveraged 2D models was not applicable to supercritical bend flows. For flows with low Fro, the vertical acceleration might be ignored; however, as Fro increased, it became significant within the bend, and its negligence led to large errors in computations. In flows with high Fro, the maximum vertical acceleration occurred at the beginning of the bend (minimum depth point), and the minimum occurred at the wave crest. At high Fro, the vertical acceleration was downward, causing the hydrodynamic pressure to become less than the corresponding hydrostatic pressure.Conclusions: The threedimensional model of Flow3D is a suitable tool for the simulation of highvelocity supercritical flows in bends in comparison with the twodimensional depthaveraged model of shallow water equation of Roe2D. By examining the pressure distribution and vertical acceleration in numerical models, it may be concluded that the basic assumption in the extraction of shallow water equations, namely the hydrostatic pressure distribution, is not admissible, especially at high Froude numbers. Moreover, the effect of vertical acceleration of water particles has a great effect on the estimation of wave crest depth and its position in the bend. .