An analytical model for vortex at vertical intakes

In the present paper, free surface vortex formation at intakes is investigated analytically. By assuming a spiral form for vortex streamlines, continuity and momentum equations were integrated and solved in a vortex flow domain. From this solution, velocity and pressure distributions were found above the intake under vortex action. An equation for the water surface profile was also found and compared with another research. By considering that in an air core vortex, pressure at the intake entrance drops to zero, a relationship was found for critical submerged depth and verified by experimental data and another analytical equation. It was concluded that the results of the proposed spiral analytical model had good agreement with the experimental data.

• Analytical analysis of free surface vortex phenomenon over a vertical intake in the reservoir. • Solving 3D Navier-Stokes equations analytically for a swirling flow. • Predicting the critical submergence at a vertical intake.
• Extracting pressures and velocities analytically using spiral theory. • Comparing obtained analytical results with previous experimental and analytical results.

NOTATIONS
The following symbols are used in this paper: D intake pipe diameter Fr intake Froude Number ¼ V/√gD Γ vortex strength N Γ circulation number g gravitational acceleration g r gravitational acceleration in r direction g z gravitational acceleration in z direction g θ gravitational acceleration in θ direction Re intake Reynolds number ¼ VD/q S intake submerged depth S c critical intake submerged depth V intake flow velocity q kinematic viscosity ρ density of water σ surface tension of water r, θ and z coordinate directions r distance from the vortex center r m radius at the maximum tangential velocity V θ tangential velocity V r radial velocity V z velocity in the z direction P pressure ω Angular velocity

INTRODUCTION
Vortices are often experienced in nature and industries; for example in the course of water withdrawing at intakes (Khanarmuei et al. 2016;Khadem Rabe et al. 2018;Tahershamsi et al. 2018;Pakdel et al. 2020) and may be important for the desalination and treatment of saline wastewater (Panagopoulos et al. 2019;Panagopoulos & Haralambous 2020;Panagopoulos 2021). In spite of their common occurrence, vortex structure, formation and dynamics, are still not completely understood (Stepanyants & Yeoh 2008). Many factors affect on vortex formation and flow field near the intake structure such as asymmetry in the inflow and reservoir geometry (Roshan et al. 2009;Amiri et al. 2011;Sarkardeh et al. 2012;Sarkardeh et al. 2013), flow velocity in the intake, submerged depth and diameter of the intake pipe. Study of free surface vortices is important and necessary not only from a purely academic view, but also for a better understanding of such fluid phenomena in practical engineering systems. Most vortices in nature, however, have a 'spatial' structure; that is, the streamlines are not perpendicular to the axis of rotation (Lugt 1983). There are visual evidences as well as computational simulations that show the vortex streamlines follow a spiral path (Lugt 1983;Shukla & Kshirsagar 2008;Lucino et al. 2010;Sarkardeh et al. 2014). A schematic and real spiral vortex are presented in Figure 1(a) and 1(b) respectively. One important case in engineering practice is withdrawing water at intakes. Surface vortex formation at intakes is an undesirable phenomenon, causing air entrainment into the intake, vibration and energy loss (Knauss 1987). Vortices can be divided into six types based on visual classification at the Alden Research Laboratory (Padmanabhan & Hecker 1984). Type 1 is observed as a weak rotation of flow at the water surface. In Type 2, in addition to water surface rotation, a dimple is also observed at the water surface. In Type 3, the rotation of the flow extends down to the intake itself. In Type 4, debris is dragged into the intake. In Type 5, some air bubbles are entrained from the water surface and are transported down to the intake. In the strongest surface vortex (Type 6), a stable air core is formed in the center of the vortex and air is steadily entrained into the tunnel (Knauss 1987). To prevent air entrainment of surface vortices, a minimum submerged depth of the intake, called the critical submerged depth S c , is recommended for the intake (Figure 2(a)). Submerged depth is defined as the vertical distance between the water surface and the level of the intake center (Figure 2(b)).
Vortex phenomenon is studied with three different possible approaches: i-experimentally, ii-numerically and iii-analytically. In the experimental studies, a physical model has been constructed so that the scale effects were minimized. Vortices were then studied in different hydraulic and geometric conditions and related hydraulic factors were measured. By collecting data, empirical equations have been developed and proposed for practical hydraulic designs such as for critical submerged depth (Denny & Young 1957;Berge 1966;Gordon 1970;Reddy & Pickford 1972;Amphlett 1976;Chang 1977;Anwar et al. 1978;Jain et al. 1978;Sarkardeh et al. 2010;Amiri et al. 2011). In numerical studies, by discretizing governing equations, flow motion equations have been solved for the flow domain. Results also have been verified by experimental data. The results of this approach could help future simulations and prevent extra costs in physical modeling (Constantinescu & Patel 1998, 2000Marghzar et al. 2003;Shukla & Kshirsagar 2008;Lucino et al. 2010;Sarkardeh et al. 2014). There have been also several attempts at analytical solution of vortex at intakes (Rott 1958;Lundgren 1985;Miles 1998;Andersen et al. 2003Andersen et al. , 2006Lautrup 2005;Stepanyants & Yeoh 2008;Yildirim et al. 2012;Schneider et al. 2015).
Maybe the first analytical study on vortex was done by Rankine (1858). He suggested a simple model for analyzing an intake vortex. In his proposed model the inner core is assumed as a forced and the outer region as a free vortex. The boundary between the forced inner and irrotational outer free vortex is located at a distance equal to the radius of the intake from the axis of rotation. In the Rankine (1858) Model the velocity component in z direction is ignored and streamlines are considered as concentric circles. Rankine (1858) also defined vortex strength by its circulation as (White 2003): where Γ is vortex strength, ω is angular velocity and V θ is the tangential velocity at a distance r from the vortex axis. It is obvious that Γ is constant in the irrotational part of the vortex and ω is constant in the forced part of the vortex. Another analytical attempt to describe the vortex phenomenon and calculate the critical submerged depth was made by Odgaard (1986). He suggested a model which provided a reasonable relationship between the critical submerged depth (S c see in Figure 2(a)) and flow parameters such as discharge, circulation, viscosity and surface tension. Odgaard (1986) model was based on exact solution of governing equations for viscous unbounded fluid. Odgaard (1986) assumed the flow situation as steady, axi-symmetric, incompressible and the radial velocity component nearly linear near the pipe entrance. He also, by integrating equations of motion and Navier-Stokes, developed the following equation without the surface tension term for calculating critical submerged depth: Many simplifying assumptions were made to derive Odgaard (1986) equations.
In the present paper, by applying spiral form for vortex streamlines in continuity and momentum equations, governing equations in complete form were integrated and 3D flow domain was solved in the presence of an air core vortex. The critical submerged depth was then determined by implementing independent non-dimensional parameters. Moreover, the water surface and formed air core were plotted regarding the extracted equations from the spiral analytical model. Finally, proposed equations were compared with other researcher results.

Corrected Proof THEORY AND GOVERNING EQUATIONS
The governing continuity and Navier-Stokes equations describing flow in a steady state incompressible fluid are given by the following set of equations (White 2003): where V r , V z and V θ are the radial, axial and tangential components of the velocity field; g is the acceleration due to the gravity; ρ is the fluid density; p is the pressure; and q is the kinematic viscosity. A spiral path in space in cylindrical coordinate scan be described by the following equations: where a, b and c are the controller coefficients to produce different spiral curves. Few spirals are plotted in Figure 3 using Equations (8)-(10).

Free vortex region
Applying Equation (1) in Equations (11)-(13), and using Equations (8)-(10), tangential, axial and radial velocity components can be written as: One can plot spiral streamlines and velocity vectors from Equations (17) to (19) (Figure 4). For example by assuming Γ ¼ 0.4 and θ between zero and 2π, three streamlines are plotted in Figure 4(a). 2D plot of these streamlines in xy plane is shown in Figure 4(b) and vector plot of xy component of velocity vector is also plotted in Figure 4(c).

Forced vortex region
At forced vortex region by applying Equation (2) in Equations (11)-(13), and using Equations (8)-(10), tangential, axial and radial velocity components can be written as: Spiral streamlines and velocity vectors were plotted from Equations (29) to (31) in Figure 5 as well as assumptions for plotting Figure 4.

Corrected Proof
Integrating Navier-Stokes equations for pressure distribution at forced vortex region: The origin of the cylindrical coordinate system (r, z, θ) was considered at the reservoir surface, and the z-axis directed downwards ( Figure 2). Substituting Equations (29)-(31) in Equations (5)-(7) and integrating them yields: In r direction: In u direction: In z direction: where C 1 , C 2 and C 3 are constants. By considering: Therefore: where P r , P θ and P z are the integrated pressure equations along the r, θ and z respectively. Finally, pressure distribution is: where C equals to zero in case of no vortex (ω ¼ 0 → P ¼ ρgz).

FLUID SURFACE PROFILE
The surface profile of a vortex derived through considering P equal to zero at Equations (28) and (40) for free and forced vortex regions respectively and neglecting viscosity due to its small effect (Hite & Mih 1994): where H is the water surface, C is the constant coefficient, a 1 is 3Γ 2 /θ 2 r 4 and a 2 is (9θ 2 Γ 2 þ 3Γ 2 )/θ 2 r 2 . For example in Figure 6, Equations (41)a and (41)b is plotted for Γ ¼ 0.4, ω ¼ 2.5, g ¼ 9.81, π ¼ 3.14 and C ¼ -0.081. Corrected Proof Equation (45) has one real and two unreal roots and can be analytically solved for S/D. It should be noted that to calculate the critical submergence in the present analytical model, the boundary between force and free vortex region at the intake entrance was considered equal to D/4. To verify Equation (45), experimental results of Paul (Odgaard 1986) and analytical equation of Odgaard (1986) was utilized. Paul (Odgaard 1986) measured critical submerged depth at a vertical intake ( Table 1).
The prediction of the present spiral model (Equation (45)) for critical submerged depth is compared with experimental data of Paul (Odgaard 1986) in Figure 8. Results of Odgaard (1986) equation (Equation (3)) are also shown in this figure.
As can be seen from Figure 8, Odgaard (1986) over estimated S/D with a different trend compared to the experimental data. Figure 8 shows that the results of the present analytical study are near to the experimental data and with a similar trend.

CONCLUSIONS
In the present analytical study, a spiral model for vortex flow was introduced. In this model, the spiral equations were applied in the 3D cylindrical Navier-Stokes equations. After substituting, differentiating and integrating equations, velocity and pressure distributions and also fluid surface profile were yielded. For a practical verification, an air core vortex as a critical state at intakes was selected. Water surface of an air core free surface vortex was plotted and compared with Hite & Mih (1994) proposed equation and the deviation was found to be about 20% in the maximum point. Using the present spiral model, an equation for critical submerged depth was derived and compared with experimental data of Paul (Odgaard 1986) and analytical equation of Odgaard (1986). Results showed that present spiral model could predict (S/D) cr in good agreement with experimental data.

DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.  (Odgaard 1986) and analytical model of Odgaard (1986).