As this investigation is based on numerical approximation, the Spectral method is considered a good approach to solving the Eqs. (7)–(9) (Gottlieb et al. [26]). Here, in a series, the variables will be expanded including Chebyshev polynomials, i.e., φn(x) and ψn(x) are the expanded functions can be narrated as (12)ϕn(x)=(1−x2)Pn(x),ψn(x)=(1−x2)2Pn(x)}where Pn(x)=cos⁡(ncos−1⁡(x)) represents the Chebyshev polynomial of nth order. What is more, φn(x) and ψn(x) are applied to expand Ω(x,y,t),Ψ(x,y,t) and τ(x,y,t) as (13)Ω(x,y,t)=∑m=0M∑n=0NΩmn(t)ϕm(x)ϕn(y)Ψ(x,y,t)=∑m=0M∑n=0NΨmn(t)ψm(x)ψn(y)τ(x,y,t)=∑m=0M∑n=0Nτmnϕm(x)ϕn(y)+x}

The collocation points (xi,yj) are taken as (14)xi=cos⁡[π(1−iM+2)],i=1,…,M+1yj=cos⁡[π(1−jN+2)],j=1,…,N+1}

Steady solutions are obtained by the Newton-Raphson iteration method as (15)Ω(κ+1)=C1−1N1(Ωmn(κ),Ψmn(κ)),τmn(κ)),Ψ(κ+1)=C2−1N2(Ωmn(κ),Ψmn(κ)),τmn(κ)),τ(κ+1)=C3−1N3(Ωmn(κ),Ψmn(κ)),τmn(κ)),}where κ is the iteration number. The assurance of convergence is instituted as small as possible for εκ(εκ < 10−10) defined as (16)εκ=∑m=0M∑n=0N[(Ωmn(κ+1)−Ωmnκ)2+(Ψmn(κ+1)−Ψmnκ)2+(τmnκ+1−τmnκ)2]

At last, to get the solution of UF, Crank-Nicolson (C-N formula) and Adams-Bashforth (A-B method) strategies are applied to the Eqs. (7)–(9).

The definition of flow resistance coefficient (λ) is expressed as (17)P1∗−P21∗Δz∗=λdh∗12ρ⟨Ω∗⟩2

The mean axial velocity ⟨Ω∗⟩ is enumerated by (18)⟨Ω∗⟩=v42δd∫−11dx∫−11Ω(x,y,t)dy

Since (P1∗−P2∗)/Δz∗=G,λ is related to the mean non-dimensional axial velocity ⟨Ω⟩ as (19)λ=162δDn3⟨Ω⟩2

According to Mondal et al. [27], the Nusselt number to find out the heat transfer in steady solution branches can be expressed as (20)Nuc=12∫−11(∂τ∂x)x=−1dy;Nuh=12∫−11(∂τ∂x)x=1dywhere Nuc and Nuh are denoted the Nusselt number for cooled and heated sidewalls, respectively. On the contrary, for unsteady solutions, it is defined as (21)Nuτc=12∫−11⟨⟨∂τ∂x|x=−1⟩⟩dy,Nuτh=12∫−11⟨⟨∂τ∂x|x=1⟩⟩dy

Here, the percentage errors of our present study will be justified while executing numerical values by the grid efficiency method since the effectiveness of the algorithm for the governing equations can be supported by this significant method. The necessity of grid accuracy was realized by Wang et al. [28] when they visualized the variation of bifurcation structure with the variation of grid points. Though Dolon et al. [21] investigated the accuracy of the numerical algorithm, the authors did not find any relative percentage error. In the present study, the grid efficiency is represented by relative percentage error for both R-C (λ) and axial velocity (Ω) using the fixed value of Dn=1000, Gr=1000, Pr=7.0, δ=0.001, and aspect ratio Ar = 2.0 for various grid sizes as 14 × 28, 16 × 32, 18 × 36, 20 × 40, 22 × 44, 24 × 48, 26 × 52. Our obtained percentage errors for both λ and Ω are presented in Table 1 and are shown graphically in Fig. 2. The formula that is used to find out the percentage error is

εp=|currentvalue−previousvaluecurrentvalue|×100%

It is seen from Table 1 that εp(λ) and εp(Ω) changes to 0.4822075231% and 1.1005607599% respectively when the grid points are changed from 14×28 to 16×32. Both εp(λ) and εp(Ω) are reduced monotonically as the grid points are increased. For 20 × 40 when these grid sizes are altered from 18 × 36, then εp(λ) and εp(Ω) are figured out as 0.0812972035% and 0.3362469682% consequently. With the increment of grid sizes, more accuracy is obtained which is noticed in Fig. 2. Considering all these aspects, M = 20, and N = 40 grid sizes are used in our present study to execute the numerical solutions because substantial accuracy can be obtained using this grid size.

This section provides a detailed validation of our numerical outcomes by comparing them with experimental results to confirm the accuracy of the secondary flow structures and temperature distribution. The experimental vortex structures were observed by Chandratilleke [29,30] and Sugiyama et al. [31] for a curved rectangular duct (CRD) with an aspect ratio of Ar = 2.0. As displayed in Fig. 3, the secondary flow structures on the right are the current numerical results while the left displays the experimental results by Chandratilleke [29] (Fig. 3a) and Sugiyama et al. [31] (Fig. 3b). Furthermore, on the left in Fig. 3c,d. Chandratilleke [29] found the temperature field of outside heated CRD by using CFD tools. Chandratilleke [29,30] focused on the secondary flow characteristics and convective heat transfer in a CRD with external heating, revealing the formation of secondary vortices and their impact on heat transfer efficiency. Sugiyama et al. [31] visualized the secondary flow in curved rectangular channels providing insights into the flow patterns and vortex structures.

Using CFD tools, Chandratilleke et al. [32] analyzed the temperature field in an externally heated CRD highlighting the thermal behavior and heat transfer mechanisms. Our numerical simulations, conducted using a spectral-based numerical approach, accurately captured the secondary flow patterns and vortex structures observed in the experiments. The comparison of numerical and experimental results, shown in Fig. 3, where the left side displays the experimental results and the right side shows the numerical results, confirms the validity of our numerical approach. The temperature distribution obtained from the numerical simulations also showed a high degree of agreement with the experimental CFD results by Chandratilleke et al. [32] demonstrating the reliability of the numerical method in predicting thermal behavior.

After a comprehensive exploration, we have procured four branches of steady solutions (SS) in the fluid while flowing through the RCD. The bifurcation diagram in Fig. 4a shows the relationship between the Dean number and the flow resistance coefficient (λ). The diagram reveals multiple branches of steady solutions, each corresponding to different flow patterns. The first branch, which spans the entire range of the Dean numbers, represents the primary flow structure with a single vortex. The second, third, and fourth branches correspond to more complex flow patterns with multiple vortices. These branches exhibit various bifurcation points, where the flow transitions from one steady-state solution to another. The illustrated figure is so perplexing and that’s why we have drawn an enlargement of the entangled part for a larger Dn in Fig. 4b.

The branching structure of steady solutions involves understanding the bifurcation phenomena in fluid dynamics. Bifurcation occurs when a small change in the system’s parameters, such as the Dean number, causes a sudden qualitative change in its behavior. In curved ducts, the centrifugal force induced by the curvature leads to the formation of secondary flows, which can bifurcate into multiple steady-state solutions. These solutions represent different flow patterns, characterized by the number and arrangement of vortices. The stability of these solutions depends on the balance between the centrifugal force, pressure gradient, and thermal effects. As the Dean number increases, the flow transitions through various bifurcation points, leading to the complex branching structure observed in the steady solutions.

The 1st SS branch ccupying the whole range of Dn is depicted in Fig. 5a for 10≤Dn≤3000. This branch starts at point a (Dn ≈ 0) and prolongs with the rising Dn and decreasing λ up to point b(Dn≈208.17384). It then turns back to point c (Dn≈210.593) and the branch finally reaches point d (Dn≈3000) without further turning. Fig. 5b is the enlargement of Fig. 5a at points b and c which indicates very smooth turning. The axial velocity distribution (AVD), secondary flow pattern (SFP), and energy profiles at various Dn in this branch are shown in Fig. 5e,f. At Dn=10, the axial velocity exists in the central line while turning to the left, i.e., near the outer boundary of the duct with high velocity at Dn=3000 which is justified by Fig. 5c,d, respectively because frequencies of path lines at Dn=3000 are higher than that of Dn=10. It is transparent that the velocity, however, with the growth of Dn, increases on the boundary, and the maximum velocity is marked up at Dn=3000. The first branch consists of a single-vortex to a 3-vortex solution which starts with a single-vortex solution with low pressure and ends as an asymmetric 2-vortex with high pressure. The single vortex remains up to Dn≈102.593 and then a new vortex is created on the outer wall of the duct because of heating-induced buoyancy force. The 2nd SS branch between Dn=611 and Dn=3000 is illustrated in Fig. 6a. Though the lower part of the branch looks quite simple starting at point a (Dn=3000) to point b (Dn=611.667), many complexities can be noticed with some smooth and sharp turnings between the ranges 2000≤Dn≤2350 and 2400≤Dn≤2850 (see Fig. 6c,d).

The point b is very smooth as shown in enlarged Fig. 6b. The AVD and SFP for Dn = 611 and Dn = 3000 concerning grid points are shown in Fig. 6e,f, respectively. Exploring the velocity distribution, the changes are visible as compared to the 1st steady branch. Both the higher-pressure gradient and external heat are responsible for increasing the velocity. The 3rd SS branch occupying the range 1027≤Dn≤3000 is portrayed in Fig. 7a. The branch starts at point a for Dn = 3000 (λ = 0.00802) and finally reaches the point k (Dn = 3000, λ = 0.00887) amidst many turnings that are shown with appropriate zoom Fig. 7b–d for the ranges 1027≤Dn≤3000, 1700≤Dn≤2050, and 2700≤Dn≤2850, respectively. AVD and SFP for Dn=1027 and Dn=3000 concerning grid points are shown in Fig. 7e,f, respectively, and can be observed the presence of a high-pressure gradient and the impact of external heat.

The fourth SS branch is located in the region of 801≤Dn≤3000 which commences from Dn=3000 (point a) and finally reaches point j (Dn = 3000, λ = 0.00876), and is displayed in Fig. 8a. Then extends for increasing λ and decreasing Dn up to point b (Dn = 805.2, λ = 0.01496) with two strict turnings at Dn = 801.95 (λ = 0.01494) and Dn = 813.83 (λ = 0.01479). From here, the branch then prolongs in reverse direction and gets to point c (Dn = 1085.25, λ = 0.01354) followed by point d (Dn = 1006.12, λ = 0.01399). After reaching point e (Dn = 1578.01, λ = 0.01157), the branch becomes more intertwined between point f (Dn = 1499.84, λ = 0.01197) and g (Dn = 1603.55, λ = 0.01148) (see Fig. 8c). Fig. 8d shows the transition from point h by a strict turning tends to point i. The distribution of axial velocity and secondary velocity at Dn = 801 and Dn=3000 concerning grid points is shown in Fig. 8e,f, respectively. Typical contours of SF are drawn in Fig. 9 for various Dean numbers where it can be explored that the branch starts with asymmetric 2-vortex solutions at Dn = 3000 and remained 2-vortex up to Dn≈1128.27169. The whole branch consists of asymmetric 2- to 12-vortex solutions at various Dn’s.

The stability of the steady solutions is influenced by the thermal effects, particularly the heating-induced buoyancy force. In the present study, the bottom and outer walls of the duct are heated, while the upper and inner walls are kept at room temperature. This temperature gradient creates a buoyancy force that interacts with the centrifugal force, further complicating the flow dynamics. The heating-induced buoyancy force can either stabilize or destabilize the flow, depending on the balance between the thermal and centrifugal effects.

Here, we have analyzed the vortex structure and the number of secondary vortices at a specific Dn’s throughout every SS branch as shown in Figs. 9 and 10, respectively, which means how the vortices looked like in each branch at a fixed Dn and how many vortices have arisen. The formation of secondary vortices is driven by the interaction between the centrifugal force and the pressure gradient. As the fluid flows through the curved duct, the centrifugal force pushes the fluid toward the outer wall, creating a pressure gradient across the duct. This pressure gradient induces secondary flows, which manifest as vortices. The number and arrangement of these vortices depend on the Dean number and the thermal effects. At higher Dean numbers, the flow becomes more unstable, leading to the formation of multiple vortices and complex flow patterns. At first, we investigated for Dn = 500, where we observed that the only first SS branch contains this value and obtained 2-vortex at Dn = 500. The next investigated value is Dn = 1000 where it can be seen that a 2-vortex structure exists in the 1st and 3rd branches while 7 vortices exist in the 2nd and 4th branches. Then, we found 2, 8, and 11 vortices for Dn = 1500 for the consecutive branches. The number of vortices in the 1st steady branch remained unchanged up to Dn = 2000 whereas decreased by 2-vortices in the 2nd branch but one more vortex was created in the 3rd and 4th branches. The same number of vortices are attained in the 1st and 3rd branches, is declined to 5-vortex in the 2nd, and is increased by one in the 4th branch for Dn = 2500. Except for the third branch (the highest 12-vortex are found), the other three branches have continued with the identical number of vortices for Dn = 3000.

This section delves into the non-linear characteristics of unsteady flow (UF) through a loosely bent rectangular duct by examining the time evolution of the flow resistance coefficient (λ) across a range of Dn. The analysis reveals various flow regimes, including steady-state, periodic, multi-periodic, and chaotic flows, each characterized by distinct vortex structures and flow patterns.

At a low Dean number (Dn = 10), the flow remains in a steady state, as shown in Fig. 11a. The axial velocity distribution (AVD) and secondary velocity distribution (SVD) for Dn = 10 are depicted in Fig. 11b. The AVD is directed towards the center of the duct, while the SVD decreases significantly when crossing grid number 450. This behavior is attributed to the low-pressure gradient and centrifugal force. The flow pattern, illustrated in Fig. 11c, consists of an asymmetric single-vortex solution. The steady-state flow persists up to Dn = 86, with the flow resistance coefficient (λ) decreasing to 0.06065. As the Dean number increases slightly beyond 86, a notable shift occurs in the UF pattern. At Dn = 86.5, the steady-state flow transitions to a periodic flow, as shown in Fig. 12a. The periodic oscillation is traced between specific time intervals, confirmed by the power spectrum density (P-S) in Fig. 12b. The grid-point-wise distribution of axial and secondary velocities for Dn = 86.5 is presented in Fig. 12c, showing increased velocity distributions due to the pressure gradient. The flow pattern for a single period (14.88≤t≤15.94) of oscillation is depicted in Fig. 12d, consisting of asymmetric 2-vortex solutions, with the upper vortex dominating over the smaller vortex from the lower wall of the duct.

The periodic flow becomes multi-periodic (M-P) for Dn = 202 as shown in Fig. 13a in the t−λ plane. The flow pulsates between λ=0.0328 and λ=0.0389. To clear the obscurity of whether the flow is M-P or not, we draw the P-S of this TDF as shown in Fig. 13b. The elementary frequencies, their harmonics, and more lines spectrum confirm to us, that the flow is M-P. For Dn = 202 and at t = 17.32, the grid-point-wise AV and SV are sketched in Fig. 13c. The contours of axial flow (AF), SF, and isotherms for this M-P flow are presented in Fig. 13d for a single period of oscillation 9.09≤t≤17.32. It is seen that the SF is composed of asymmetric 2- and 3-vortex solutions. Due to the increase in temperature and pressure gradient in the lower and outer walls, two new vortices are generated; one from the lower wall and the other from the outer wall of the duct.

We have studied the UF precisely to explore all possible solutions. It is interesting to see that the M-P flow again turns periodic for Dn = 400 with the period between λ=0.02315 and λ=0.02352 as depicted in Fig. 14a. Fig. 14b describes the P-S where it is seen that the flow is periodic. The AVD as explained grid-point-wise (reporting the SV in the same figure always) is depicted in Fig. 14c at t = 14.67. The flow patterns are presented in Fig. 14d, where it is exposed that the periodic flow is composed of an asymmetric 2-vortex solution. After a slight increase in the pressure gradient at Dn = 430, the flow transitions back to M-P, oscillating between specific time intervals with asymmetric 2- and 3-vortex solutions, as shown in Fig. 15a,d. The P-S in Fig. 15b confirms the M-P nature of the flow. The grid-point-wise AVD and SVD for Dn = 430 are illustrated in Fig. 15c, showing increased fluid velocity and temperature due to the influence of the outer wall of the duct. To visualize the fluctuating behavior of the unsteady solutions again, it is drawn for Dn = 500 as in Fig. 16a, where it is seen that the flow is chaotic with pulsates between λ=0.0208 and λ=0.0212. To eliminate our doubt about whether it is periodic or chaotic, we exhibit P-S of TDF (see Fig. 16b). The flow can be defined as transitional chaos (Mondal et al. [27]). Also, the grid-point-wise velocity distribution of the axial flow and secondary velocity is depicted in Fig. 16c at t = 15.26. For a specific time interval, 15.26≤t≤16.33, the pattern fluctuation of AF, SF, and isotherms is shown in Fig. 16d, from which it can be seen that the SF is composed of asymmetric 4-vortex solutions. As much as increasing the pressure gradient for example Dn = 3000, the flow is remained unchanged and becomes more chaotic with 5- to 10-vortex solutions, which is termed strong chaos (Mondal et al. [27]) which means we have found no significant change except the strongest chaotic flow as depicted in Fig. 17a,d. The P-S of TDF and the grid-point-wise velocity distribution are displayed in Fig. 17b,c. Therefore, all the unsteady solutions are summarized by illustrating them as a percentage as well as vortex number. In the current investigation range of Dean number, 0<Dn≤3000, the region of the steady-state flow is 0<Dn≤86,the periodic/multi-periodic flow interval is 86.5≤Dn≤430, and the rest of the region is for the chaotic flow with some transitions. The schematic representation of the percentage with vortex number is demonstrated in Fig. 18. From the figure, we observed that the maximum area holds the chaotic flow of approximately 86% with 2- to 10-vortex solutions while the steady-state flow holds a minimum area of approximately 3% with single and 2-vortex solutions and periodic flow consists of 2- and 3-vortex solution with 11% of the total flow region.

This time advancement flow analysis reveals the complex dynamics of unsteady flow in a curved rectangular duct, driven by the interplay between centrifugal force, pressure gradient, and thermal effects. The Dean number is crucial in determining the flow regime and stability of flow patterns. At low Dn’s, the flow remains stable with a single vortex structure due to the low-pressure gradient and centrifugal force. As the Dn increases, the flow transitions to a periodic regime characterized by oscillations and asymmetric 2-vortex solutions influenced by pressure gradient and thermal effects. Further increase in the Dean number leads to multi-periodic flow with more complex oscillations and multiple vortex structures, resulting from the interaction between centrifugal force and thermal effects. At higher Dn’s, the flow becomes chaotic with irregular oscillations and complex vortex structures driven by strong centrifugal force and thermal effects, leading to multiple vortices and flow instabilities. By remembering these transitions and dynamics, one can substantially optimize the fluid flow and heat transfer in engineering applications involving curved ducts.