Modeling of scour hole characteristics under turbulent wall jets using machine learning

Considering all the mandatory information for the evaluation, each a number of linear regression evaluation (MLRA) and non-linear a number of regression evaluation (MLRA) are carried out. The nondimensional dependent variables thought of on this examine are most equilibrium scour depth (left({d}_{s}/aright)), the gap to scour depth from the top of inflexible apron (left({x}_{0}/aright)), the peak of dune (left({d}_{d}/aright)) and the gap to most peak of dune crest from the top of inflexible apron (left({x}_{d}/aright)). From Gamma check, densimetric Froude quantity (Fd), apron size (L), tail water stage (dt), median sediment dimension (D50) are discovered to be influencing these 4 dependent parameters. For modelling, apron size (L), tail water stage (dt) and median sediment dimension (D50) are made dimensionless numbers by dividing them with peak of gate opening (a) equivalent to L/a, dt/a, D50/a respectively. The outcomes of variation of dependent parameters with enter parameters are analyzed as demonstrated in Fig. 4a–d. Present analysis reviews a rising development between all dependent parameters in opposition to the apron size (L/a), as proven in Fig. 4a. The motive for this development is attributable to the dissipation of vitality of the jet because it travels over the apron. Hence, longer aprons are in a position to dissipate bigger vitality and cut back the erosive capability of the jet. Similarly, rising traits are additionally seen for the variations of dependent parameters with tail water stage (Dt/a), densimetric Froude quantity (Fd), and median sediment dimension (D50), as proven in Fig. 4.Figure 4Relationship of impartial and dependent parameters (ds/a, × 0/a, xd/a, dd/a) (a) L/a, (b) dt/a, (c) D50/a, (d) Fd.A quantity of single regressions fashions for max equilibrium scour depth (left({d}_{s}/aright)), the gap to scour depth from the top of inflexible apron (left({x}_{0}/aright)), the peak of dune (left({d}_{d}/aright)) and the gap to most peak of dune crest from the top of inflexible apron (left({x}_{d}/aright)) with totally different enter parameters are established. After rigorous examine, one of the best fashions for every couple (dependent vs impartial) with excessive coefficient of willpower R2 are recognized. Then, a number of regression evaluation has been carried out and two equations (one for linear and one other for nonlinear circumstances) are obtained for every dependent parameter, as supplied beneath.Resulted equations from a number of linear regression evaluation (MLRA)MLRA for max equilibrium scour depth (left({d}_{s}/aright))$$frac{{d}_{s}}{a}=0.293left({d}_{t}/aright)-0.055left(L/aright)+7.489left({d}_{50}/aright)+0.270{F}_{d}-1.437,$$
(8)
MLRA for distance to scour depth from the top of inflexible apron (left({x}_{0}/aright))$$frac{{x}_{0}}{a}=2.295left({d}_{t}/aright)-0.371left(L/aright)+58.996left({d}_{50}/aright)+1.807{F}_{d}-12.317,$$
(9)
MLRA for distance to most peak of dune crest from the top of inflexible apron (left({x}_{d}/aright))$$frac{{x}_{d}}{a}=2.61left({d}_{t}/aright)-0.433left(L/aright)+83.677left({d}_{50}/aright)+2.204{F}_{d}-13.661.$$
(10)
MLRA for the peak of dune (left({d}_{d}/aright)).$$frac{{d}_{d}}{a}=0.279left({d}_{t}/aright)-0.062left(L/aright)+1.476left({d}_{50}/aright)+0.071{F}_{d}-0.573.$$
(11)
Resulted equations from a number of non-linear regression evaluation (MNLRA)MNLRA for max equilibrium scour depth (left({d}_{s}/aright))$$frac{{d}_{s}}{a}=6.395 {left(frac{L}{a}proper)}^{-1.378}{left(frac{{d}_{t}}{a}proper)}^{0.90}{left(frac{{D}_{50}}{a}proper)}^{0.465}{left({F}_{d}proper)}^{1.469}.$$
(12)
MNLRA for distance to scour depth of from the top of inflexible apron (left({x}_{0}/aright))$$frac{{x}_{0}}{a}=36.03 {left(frac{L}{a}proper)}^{-1.199}{left(frac{{d}_{t}}{a}proper)}^{0.828}{left(frac{{D}_{50}}{a}proper)}^{0.473}{left({F}_{d}proper)}^{1.393}.$$
(13)
MNLRA for the gap to most peak of dune crest from the top of inflexible apron (left({x}_{d}/aright))$$frac{{x}_{d}}{a}=40.94 {left(frac{L}{a}proper)}^{-1.04}{left(frac{{d}_{t}}{a}proper)}^{0.753}{left(frac{{D}_{50}}{a}proper)}^{0.466}{left({F}_{d}proper)}^{1.258}.$$
(14)
MNLRA for the peak of dune (left({d}_{d}/aright))$$frac{{d}_{d}}{a}=18.395 {left(frac{L}{a}proper)}^{-1.534}{left(frac{{d}_{t}}{a}proper)}^{1.08}{left(frac{{D}_{50}}{a}proper)}^{0.703}{left({F}_{d}proper)}^{1.022}.$$
(15)
The coefficient of willpower R2 for a number of linear regression equation and a number of nonlinear regression equation are discovered to be 0.56 for (frac{{d}_{s}}{a}), 0.66 for (frac{{x}_{0}}{a}), 0.67 for (frac{{x}_{d}}{a}), 0.56 for (frac{{d}_{d}}{a}) and 0.66 for (frac{{d}_{s}}{a}), 0.74 for (frac{{x}_{0}}{a}), 0.74 for (frac{{x}_{d}}{a}), 0.54 for (frac{{d}_{d}}{a}) respectively which measures the share of how a lot of the full variance is defined by the impartial variables. Further, an try has been made to use two machine learning approaches equivalent to ANN-PSO and GEP to mannequin these 4 parameters ({d}_{s}/a), ({x}_{0}/a), ({d}_{d}/a) and ({x}_{d}/a).Model growth using ANN-PSOIn this ANN-PSO modelling, a number of trials had been carried out and the coefficients C1 and C2 had been mounted at 1 and a pair of.5, 2 and a pair of.5, 1.5 and a pair of.5, 1.5 and a pair of.5 for ({d}_{s}/a), ({x}_{0}/a), ({d}_{d}/a) and ({x}_{d}/a) respectively. The error evaluation outcomes for the coaching information, testing information, and the complete dataset for numerous swarm sizes and quantity of neurons (N) for every dependent parameter had been analysed. It was noticed that the swarm dimension will increase with the identical values of C1 and C2. While sustaining the quantity of neurons fixed, the values of R2, E, and Id lower, whereas the worth of RMSE will increase.Model growth using GEPIn this part, mannequin growth for 4 dependent parameters using the GEP method is described. By incorporating all of the 4 impartial enter parameters (L/a, dt/a, D50/a, Fd), GEP expression has been derived and GeneXpro Tools 5.0 software program bundle is used for this evaluation. Using normalized information, 4 makes an attempt have been made with the variation of chromosome quantity, health perform, and quantity of runs for modelling the wall jet scouring. Table 6 reveals the corresponding parameters of the optimized GEP mannequin.The expression bushes for fashions of ({d}_{s}/a), ({x}_{0}/a), ({d}_{d}/a) and ({x}_{d}/a) are introduced in Fig. 5a–d, respectively. In this expression tree, d0 = L/a, d1 = dt/a, d2 = D50/a and d3 = Fd. In Sub-ET 1, 2 and three (Fig. 5a), C7 and C9 are constants, and their values are 3.45 and − 5.56 respectively for mannequin of ({d}_{s}/a). In Sub-ET 1 (Fig. 5b), C2 is fixed, the worth is − 8.93 for mannequin of ({x}_{0}/a). In Sub-ET 2 and three (Fig. 5c), C4 and C7 are the constants, and their values are 2.971 and − 0.376 respectively for mannequin of ({x}_{d}/a). In Sub-ET 1 and three (Fig. 5d), C4 is fixed, the worth is − 3.114 and three.145 respectively for mannequin of ({d}_{d}/a). The equations derived from the expression bushes are introduced in Eqs. (16)–(19).Table 6 Parameters of the optimized GEP mannequin for Wall Jet scouring.Figure 5(a) Expression tree for Wall jet scouring of ({d}_{s}/a). (b) Expression tree for Wall jet scouring of ({x}_{0}/a). (c) Expression tree for Wall jet scouring of ({x}_{d}/a). (d) Expression tree for Wall jet scouring of ({d}_{d}/a).Expression for ({d}_{s}/a):$$frac{{d}_{s}}{a}=left[frac{left(frac{{d}_{t}}{a}right)}{left({e}^{frac{(L/a)}{3.45}}right)+left[left(frac{L}{a}times frac{{d}_{t}}{a}right)-frac{{d}_{t}}{a}right]}proper]+left[frac{left(frac{{d}_{t}}{a}right)}{2.187+left[left(frac{L}{a}times 3.45right)-frac{{d}_{t}}{a}right]}proper]+left[frac{left(frac{{d}_{t}}{a}right)}{{left({3.45}^frac{L}{a}right)}^{frac{{d}_{t}}{a}}+left[left(5.56+{F}_{d}right)-left(frac{{D}_{50}}{a}right)right]}proper].$$
(16)
Expression for ({x}_{0}/a):$$frac{{x}_{0}}{a}={e}^{left(frac{textual content{ln}left(frac{{d}_{t}}{a}proper)}{{e}^{frac{L}{a}^{{d}_{t}/a}}+left(frac{L}{a}-8.93right)}proper)}+{e}^{left(left({textual content{ln}}left(2times frac{{D}_{50}}{a}proper)-frac{left(frac{{d}_{t}}{a}proper)}{{F}_{d}}proper)instances frac{L}{a}proper)}+{textual content{ln}}left(frac{{d}_{t}}{a}proper).$$
(17)
Expression for ({x}_{d}/a):$$frac{{x}_{d}}{a}={textual content{ln}}left(frac{{d}_{t}}{a}proper)+left[left(frac{{left(frac{{F}_{d}}{left(frac{{D}_{50}}{a}right)}right)}^{left(frac{L}{a}+frac{{d}_{t}}{a}right)}}{{{F}_{d}}^{2.971}+frac{L}{a}}right)times left(frac{{D}_{50}}{a}right)right]+left[1times {frac{{e}^{frac{{D}_{50}}{a}}}{left(frac{L}{a}right)}}^{left(frac{{D}_{50}}{a}-0.376right)}right].$$
(18)
Expression for ({d}_{d}/a):$$frac{{d}_{d}}{a}=left[left(frac{{D}_{50}}{a}times frac{{d}_{t}}{a}right)times left(-3.114times {F}_{d}right)times 9.697times {text{ln}}left(frac{L}{a}right)right]+left(frac{{D}_{50}}{a}instances frac{{d}_{t}}{a}proper)instances {left({F}_{d}instances frac{L}{a}proper)}^{{F}_{d}}+left[left(left(left(frac{{d}_{t}}{a}times 3.145right)times {text{ln}}left(frac{L}{a}right)right)times {left(frac{{D}_{50}}{a}right)}^{2}times 3.145right)times {F}_{d}right].$$
(19)
Figure 6a–d reveals the connection between noticed and predicted values for the mannequin of ({d}_{s}/a), ({x}_{0}/a), ({x}_{d}/a) and ({d}_{d}/a) respectively. It is noticed that the expected mannequin of ANN-PSO offers good settlement with noticed values for all of the 4 fashions, whereas GEP reveals the unsatisfactory end result of the current examine.Figure 6(a) Observed vs predicted worth for all of the mannequin of ({d}_{s}/a). (b) Observed vs predicted worth for all of the mannequin of ({x}_{0}/a). (c) Observed vs predicted worth for all of the mannequin of ({x}_{d}/a). (d) Observed vs predicted worth for all of the mannequin of ({d}_{d}/a).Performance of uncertainty and reliability evaluationTo carry out a complete statistical evaluation of the proposed fashions, two indices specifically confidence interval (U95) and reliability index are computed. The statistical analysis of the current fashions, highlighting their predictive capabilities and robustness using uncertainty evaluation and reliability index, is introduced in Table 7.
Table 7 Comparison of efficiency outcomes for the uncertainty and reliability evaluation.Table 7 reveals the boldness interval (U95) and reliability index (RI) of MLRA, MNLRA, ANN-PSO and GEP in predicting ({d}_{s}/a), ({x}_{0}/a), ({x}_{d}/a) and ({d}_{d}/a). ANN-PSO mannequin represented the bottom values of confidence interval (U95), i.e., 0.383, 2.539, 2.805 and 0.268 when in comparison with MLRA (0.402, 2.604, 3.101and 0.293), MNLRA (0.415, 2.598, 3.063 and 0.301) and GEP (0.483, 28.800, 19.276 and 0.409) for predicting ({d}_{s}/a), ({x}_{0}/a), ({x}_{d}/a) and ({d}_{d}/a) respectively. Additionally, predictions of ({d}_{s}/a), ({x}_{0}/a), ({x}_{d}/a) and ({d}_{d}/a) supplied by ANN-PSO are extra dependable (RI = 0.573, 0.591, 0.576 and 0.548) when in comparison with different current fashions. Moreover, MNLRA reveals barely much less dependable (RI = 0.521, 0.545, 0.570 and 0.497) than ANN-PSO in predicting ({d}_{s}/a), ({x}_{0}/a), ({x}_{d}/a) and ({d}_{d}/a). GEP reveals wider confidence intervals (U95) and decrease relative index, indicating greater uncertainty and fewer dependable mannequin in predicting ({d}_{s}/a), ({x}_{0}/a), ({x}_{d}/a) and ({d}_{d}/a). This evaluation means that the ANN-PSO offers a extra constant and dependable mannequin for the prediction of ({d}_{s}/a), ({x}_{0}/a), ({x}_{d}/a) and ({d}_{d}/a).Statistical error evaluationThis part illustrates the efficiency of the 2 soft-computing fashions and two a number of regression fashions in predicting ({d}_{s}/a), ({x}_{0}/a), ({x}_{d}/a) and ({d}_{d}/a). To assess the power of current approaches, seven statistical indices are accounted together with two statistical indices equivalent to Root imply sq. error (RMSE) and coefficient of willpower (R2), and two relative indices, E and Id38,49,50,51,52. The error indices are computed for all the current fashions in phrases of MAE, MAPE, MSE, RMSE, R2, E and Id are depicted in Table 8.From Table 8, it’s discovered that for each (left({d}_{s}/aright)) and (left({x}_{0}/aright)), the error indices, i.e., MAE, MAPE, MSE and RMSE are much less for MLRA and MNLRA as in comparison with the ANN-PSO and GEP. But, the error indices, i.e., MAE, MAPE, MSE and RMSE are discovered to be much less for ANN-PSO as in comparison with MLRA, MNLRA and GEP for each ({x}_{d}/a) and ({d}_{d}/a). However, the R2 worth is extra in ANN-PSO mannequin for all predicting parameter values. E and Id values are additionally discovered to be near 1 for ANN-PSO fashions for all three predicting parameter values besides (left({x}_{0}/aright)). For (left({x}_{0}/aright)), E and Id values are discovered to be near 1 for MLRA mannequin. By evaluating all of the statistical parameters, ANN-PSO mannequin reveals higher end result as in comparison with the opposite introduced regression and delicate computing methods (Table 8).
Table 8 Error evaluation of totally different approaches in estimating ({d}_{s}/a), ({x}_{0}/a), ({d}_{d}/a) and ({x}_{d}/a) for wall jet scouring.

https://www.nature.com/articles/s41598-024-66291-8